J'aimerais remercier tout d'abord mes directeurs de thèse Édouard Oudet et Bozhidar Velichkov. Ils ont toujours su m'encourager, se montrer enthousiastes, et compréhensifs lorsque j'en avais besoin. Ils ont fait preuve d'une grande disponibilité, sans laquelle ma thèse aurait certainement été plus dicile. Enn, je garderai en mémoire de nombreux bons moments passés ensemble. Merci ! Je remercie chaudement Lorenzo Brasco et Jimmy Lamboley d'avoir rapporté cette thèse. Merci pour le grand soin avec lequel ils ont accompli cette tâche. Un grand merci également à Dorin Buccur et Blanche Buet d'accepter de faire partie de mon jury. Merci à Paul Barbaroux, qui m'a donné goût aux mathématiques, et m'a accompagné dans cette voie.

Merci à tout ceux qui m'ont encouragé. J'ai une pensée spéciale pour Sophie, bien sûr. iv Contents a. Obstacle problems and PDEs. We will mention some generalizations later on. But let us rst explain a few key properties on this simple model.

For any positive test function ψ ∈ C ∞ c (Ω), writing the minimality of u against a competitor u + tψ and taking the limit as t ↓ 0, one gets, in the sense of distributions, ∆u ≤ 0.

Moreover, assuming that u is continuous, this last inequality becomes an equality (the Euler-Lagrange equation of the Dirichlet energy) in the open set {u > φ} ∩ Ω. The set N := {u > φ} ∩ Ω, is called the non-contact set, while the set C := {u = φ} ∩ Ω is called the contact set.

The set Γ := ∂C ∩ Ω is the interface between the contact and non-contact sets. It is called the free boundary.

). This regularity is optimal. Indeed, in this case, we have almost everywhere in Ω, ∆u = χ C ∆φ.

In particular, the Laplacian of u cannot be expected to be continuous, even if φ is smooth.

. In a recent paper [32], Figalli, Ros-Oton and Serra proved that in dimension 2 and 3, the free boundary of the classical obstacle problem is 'generically regular', i.e for a 'generic' obstacle φ, the singular part of Γ is empty, and so Γ is a smooth (n -1)-manifold. For the elastic-plastic torsion problem introduced below, the free boundary is always regular ([37]).

By writing the minimality of u against the competitor u + t(v -u) and taking the limit as t ↓ 0, one can see that the classical obstacle problem (a.1) can be rephrased as follows

.

Let us end this section by mentioning two other famous obstacle type problems. In the thin obstacle problem, one minimizes the Dirichlet energy with Dirichlet boundary conditions as in

):

.

The medial axis of Ω, denoted by M(Ω), is dened as the set of points of Ω that have at least two closest points on the boundary ∂Ω of Ω:

This theorem basically shows that the set E m,λ is a good approximation of the λ-medial axis. More importantly, in Chapter 3 we will also use this strategy to approximate the cut locus of a manifold, which is introduced below.

The most important step in the proof of Theorem b.5, is to show that functions (v m ) m>0 are uniformly semiconcave.

 for a short proof with a quantitative bound, by Frank, Killip and Nam.

, Frank showed in the physical case that there exists a set which minimizes the quantity (P (E) + V α (E))/ |E|, thus proving that in Gamow's model, there is a nucleus with the least energy per nucleon.

, the authors studied a generalization of problem (e.1), where the perimeter term is replaced with a non-local perimeter.

Chapter 1

Large volume minimizers

We started working on some aspects of this Chapter under the supervision of Vincent Millot for our Master 2 thesis. The new results start with Section 1.4.

1.1

Abstract. This thesis is composed of two parts. In the rst part, we study a generalization of the variational problem of elastic-plastic torsion problem to manifolds. We show that in the case of manifolds, the problem is not equivalent to an obstacle type problem, contrary to the euclidean case, but we establish the equivalence when the parameter of the problem goes to innity. We show, as in the euclidean case, that the non-contact set contains the cut locus of the manifold, and converges to the latter in the Hausdor sense. Furthermore, we show that the minimizers of the problem are uniformly semiconcave. We deduce a stable approximation of the cut locus, in the spirit of the lambda medial axis of Chazal and Lieutier. We then use this result to compute numerically the cut locus of some surfaces of varied geometries.

In the second part, we study an extension of a nonlocal isoperimetric problem. More precisely, we add a connement potential to Gamow's liquid drop model for the nucleus.

We then study large volume minimizers. We show that for certain sets of parameters, large volume minimizers converge to the ball, or may even exactly be the ball. Moreover, we develop a numerical method for this variational problem. Our results conrm numerically a conjecture of Choksi and Peletier, in dimension 2: it seems that minimizers of Gamow'sliquid drop model are balls as long as they exist.

Résumé. Cette thèse comporte deux parties. Dans la première partie, nous étudions une généralisation du problème variationnel de torsion élastique-plastique à des variétés. Nous montrons que dans le cas des variétés, le problème n'est pas équivalent à un problème d'obstacle, contrairement au cas euclidien, mais nous établissons l'équivalence lorsque le paramètre du problème tend vers l'inni. Nous montrons, comme dans le cas euclidien, que l'ensemble de non contact contient le cut locus de la variété, et converge vers ce dernier au sens de Hausfor.

Nous montrons de plus que les miniseurs du problème sont uniformément semiconcaves. Nous en déduisons une approximation stable de cut locus, dans l'esprit du lambda axe médian de Chazal et Lieutier. Nous utilisons ensuite ce résultat pour calculer numériquement le cut locus de surfaces de géométries variées.

Dans la seconde partie, nous étudions une extension d'un problème isopérimétrique non local. Précisément, on adjoint un potentiel de connement au modèle de goutte liquide du noyau de Gamow. Nous étudions alors les minimiseurs de grand volume. Nous montrons que pour certains jeux de paramètres, les minimiseurs de grand volume convergent vers des boules, voire sont exactement des boules. Nous développons ensuite une méthode numérique pour ce problème variationnel. Cela permet de conrmer numériquement une conjecture de

I Elastic-plastic torsion problem, distance function and cut locus Introduction

In this thesis, we present some contributions to the eld of calculus of variations. Our results are both theoretical and numerical.

In Part I, we study an extension of the elastic-plastic torsion problem to manifolds and its relation with the cut locus, and derive a numerical method to compute the cut locus of a surface. Chapter 2 is based on the paper [START_REF] Générau | Laplacian of the distance function on the cut locus on a riemannian manifold[END_REF], to appear in the journal 'Nonlinearity'. The work in Chapter 3 and Chapter 4 was conducted in collaboration with Édouard Oudet and Bozhidar Velichkov, and are based on two papers to be submitted. For a synthetic view of this part, see the Outline of Part I.

In Part II, we study an isoperimetric problem that is a variation of Gamow's liquid drop model. It is based on the published paper [START_REF] Générau | Large Volume Minimizers of a Nonlocal Isoperimetric Problem: Theoretical and Numerical Approaches[END_REF]. The work in Chapter 2 was conducted in collaboration with Edouard Oudet. For a synthetic view of this part, please see the Outline of Part II.

We tried to make each Chapter relatively self-contained.

a

Obstacle problems

In part I, our focus will be on a variation of the elastic-plastic torsion problem, which is introduced in Section b. It is a particular example of an obstacle type variational problem.

The most classical obstacle problem is the following: given a smooth open set Ω ⊂ R n and some continuous functions φ : Ω → R (the obstacle) and g : ∂Ω → R (the boundary condition), one wants to minimize the following:

inf 1 2 Ω |∇v| 2 : v ∈ H 1 (Ω), v ≥ φ and v | ∂Ω = g . (a.1)
With reasonable assumptions on g and φ (basically to guarantee that there is at least one candidate to the variational problem), one can show that there is a minimizer u ∈ H 1 0 (Ω) by the direct method of calculus of variations, and that this minimizer is unique by the convexity of the Dirichlet energy. The graph of u typically models the resting position of an elastic membrane above Ω, attached at the height given by g on ∂Ω, and constrained to stay above an obstacle given by the graph of φ. There are many generalizations or variants of this famous problem, and many applications, such as uid ltration in a porous media, elasticplastic torsion, optimal control and nancial mathematics, see for instance [START_REF] Rodrigues | Obstacle Problems in Mathematical Physics[END_REF], [START_REF] Kinderlehrer | An Introduction to Variational Inequalities and Their Applications[END_REF], [START_REF] Friedman | Variational principles and free-boundary problems[END_REF]. The literature on the subject is vast, and it is still a major area of research in calculus of variations (a.1), but candidates are only constrained to stay above an obstacle φ on a (n-1)-hypersurface S ⊂ Ω, see [START_REF] Friedman | Variational principles and free-boundary problems[END_REF]Chapter 1,Section 11].

In parabolic obstacle problems, a time dependency is added. This is the case for instance of the Stephan problem, which models the evolution of the temperature of some ice immersed in water, see [START_REF] Rodrigues | Obstacle Problems in Mathematical Physics[END_REF]. A typical formulation would be

     u ≥ 0 in Ω, ∂ t -∆u = -1 in {u > 0} ∩ Ω, ∂ t -∆u ≥ -1
in Ω.

b Elastic-plastic torsion problem and medial axis

Let Ω be a smooth bounded open set of R n . The elastic-plastic torsion problem is the following variational problem, dened for any m > 0.

inf

Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 (b.1)
One can show that this functional has a minimizer by the direct method of calculus of variations. The minimizer is unique because we are minimizing a strictly convex functional over a convex subset of H 1 0 (Ω). Let v m be the unique minimizer of (b.1). Physically speaking, v m represents the stress function of a long bar of cross section Ω, twisted with an angle m.

The elastic-plastic torsion problem and the properties of its minimizer v m have been intensively studied in the 60s and 70s by various authors (such as Ting, Brézis, Sibony, Gerhardt, Caarelli, Rivière, Friedman, Pozzi...), see for instance [START_REF] Ting | Elastic-plastic torsion problem III[END_REF], [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF], [START_REF] Brezis | Multiplicateur de lagrange en torsion elasto-plastique[END_REF], [START_REF] Caarelli | The Smoothness of the Elastic-Plastic Free Boundary of a Twisted Bar[END_REF], [START_REF] Ting | Elastic-plastic torsion problem over multiply connected domains[END_REF], [START_REF] Caarelli | The lipschitz character of the stress tensor, when twisting an elastic plastic bar[END_REF], [START_REF] Friedman | The free boundary for elastic-plastic torsion problems[END_REF] and [START_REF] Caarelli | Unloading in the elastic-plastic torsion problem[END_REF].

An important tool in the study of this problem is the equivalence with an obstacle problem.

A consequence of the gradient constraint |∇v m | ≤ 1, is the following bound on v m :

v m ≤ d ∂Ω ,
where d ∂Ω is the distance function to the boundary of Ω:

d ∂Ω (x) = min y∈∂Ω |x -y| .
Indeed, as Ω is smooth and v m ∈ H 1 0 (Ω), we may extend v m by 0 outside Ω, to get a function

v m ∈ H 1 (R n ), such that |∇ v m | = 1 Ω |∇v m | ≤ 1. So v m is 1-Lipschitz on R n ,
and so v m is 1-Lipschitz on Ω for the euclidean metric. In particular, for any x ∈ Ω and y ∈ ∂Ω such that d ∂Ω (x) = |x -y|, we have

v m (x) ≤ v m (y) + |x -y| = |x -y| = d ∂Ω (x). So v m is also a competitor in the obstacle problem inf Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), v ≤ d ∂Ω . (b.2)
Note that the above problem also admits a unique minimizer, by the convexity of the functional and the constraint. Let us call it v d m ('d' is for the 'distance' obstacle). Brézis and Sibony showed in [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF] that v m is actually also the minimizer of this problem, i. Then, the minimizer v m of F under the constraint v m ∈ K 1 is also a minimizer of F under the constraint v m ∈ K 2 .

To prove this, one only needs to prove that ∇v d m ≤ 1. Let us give a sketch of their proof. For λ > 0 small, let v λ be the unique minimizer of

inf Ω |∇v| 2 -mv -λv 2 : v ∈ H 1 0 (Ω), v ≤ d ∂Ω . (b.
3

)
The idea is to show that |∇v λ | ≤ 1, and take the limit as λ → 0 to conclude. Let ṽ : R n → R be the extension of v λ by 0 outside Ω. For h ∈ R n , dene v + h (x) = max{ṽ(x -h) -|h| , ṽ(x)}, E + h = {x ∈ R n : ṽ(x -h) -|h| > ṽ(x)}, v - h (x) = max{ṽ(x + h) + |h| , ṽ(x)}, E - h = {x ∈ R n : ṽ(x + h) + |h| < ṽ(x)}.

Using the fact v + h and v - h are also competitors in (b.3), one can write the minimality of v λ in (b.3) against these competitors. After some computations, one gets λ E + h (ṽ(x -h) -ṽ(x) -|h|) 2 dx ≤ -

E + h
|h| (ṽ(x -h) -ṽ(x) -|h|) dx, which implies that the measure of E + h is null, and conclude the proof. Generalizing these techniques, Treu, Mironescu, Mariconda and Sifdari generalized this result to a broader class of variational problems with convex constraints on the gradient, see [START_REF] Treu | On the equivalence of two variational problems[END_REF], [START_REF] Mariconda | Gradient Maximum Principle for Minima[END_REF] and [START_REF] Safdari | The regularity of some vector-valued variational inequalities with gradient constraints[END_REF]. However, none of these will apply to our variant of the problem on manifolds, for which the equivalence of constraints fail in general (see Proposition 3.3.1).

The optimal regularity v m ∈ C 1,1 (Ω) was rst obtained in [START_REF] Caarelli | The lipschitz character of the stress tensor, when twisting an elastic plastic bar[END_REF], using the equivalent formulation (b.2). The elastic and plastic sets are dened as follows:

• Figure 1: A circle with its medial axis, and a polygonal approximation of a circle, with its medial axis.

Equivalently, M(Ω) is the set of points of Ω at which the distance function d ∂Ω is not dierentiable. In [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF], Caarelli and Friedman used the notion of ridge of Ω introduced by Ting, which is precisely the closure of M(Ω). Equivalently, it is the complement of the largest open set where the distance function is smooth.

They proved the following.

Theorem b.2 (Theorems 3.2 and 4.1 in [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF].). For any m > 0, we have

M(Ω) ⊂ E m .
Furthermore, there exists a constant C > 0 depending on Ω only such that

∀m > 0, d H (M(Ω), E m ) ≤ C m ,
where d H denotes the Hausdor distance.

The λ-medial axis. The medial axis, also called skeleton, is used in a wide variety of applications in image processing and computer vision. Many researchers have proposed a method to compute it, and are still trying to improve its computation. We refer to the book [START_REF] Saha | Chapter 1 -Skeletonization and its applications a review[END_REF] from 2017 for a review about the medial axis, its computation and applications.

One important geometric property of the medial axis M(Ω) is that it is unstable with respect to small perturbations of the boundary of Ω -see Figure 1. This instability makes computing numerically M(Ω) quite tricky. Indeed, any numerical approximation of Ω (for instance, with polygons) might introduce an articial (and large) medial set. In order to give a theoretical framework to tackle the problem of stability, Chazal and Lieutier dened in [START_REF] Chazal | The λ-medial axis[END_REF] the so called λ-medial axis of Ω, by setting for any λ > 0, M λ (Ω) := {x ∈ Ω : r(x) ≥ λ}, where r(x) is the radius of the smallest ball containing the set of all closest points to x on ∂Ω, i.e. the set {z ∈ ∂Ω : |x -z| = d ∂Ω (x)}. Note that the map λ → M λ (Ω) is non-increasing, and we have

M(Ω) = λ>0 M λ (Ω).
It is further proved in [23, Section 3, Theorem 2] that M λ (Ω) has the same homotopy type as M(Ω), for λ small enough. These facts justify that, for xed Ω, M λ (Ω) is a good approximation of M(Ω), for λ small enough. The crucial dierence though is that M λ (Ω) is stable with respect to small variations of Ω, whereas M(Ω) is not. Indeed we have the following theorem. Theorem b.3 (Consequence of Theorem 3 in [START_REF] Chazal | The λ-medial axis[END_REF]). For any 0 < λ < λ, there exists ε > 0 such that, if Ω and Ω are two open bounded subset of R n such that d H (Ω c , Ω c ) < ε, then

sup x∈M λ ( Ω) d(x, M λ (Ω)) ≤ 2 diam(Ω)ε,
where diam(Ω) is the diameter of Ω.

This means that although it is dicult to compute numerically M(Ω) directly, one can compute M λ (Ω) instead. In the same paper [START_REF] Chazal | The λ-medial axis[END_REF], the authors use Voronoi diagrams to compute an approximation of M λ (Ω) from a noisy sample of points of the boundary of Ω.

We will need the following denition. Denition b. [START_REF] Albano | On the stability of the cut locus[END_REF]. A function u : Ω → R is said to be C-semiconcave (or just semiconcave) if the function x → C |x| 2 -u(x) is convex.

As Ω is smooth, we know that d ∂Ω is semiconcave (see [22, (iii) in Proposition 2.2.2]). As explained with more details in the Appendix A.1, this implies that, as concave functions, d ∂Ω has a well dened generalized gradient whose norm is given by the formula:

|∇d ∂Ω | (x) = max(0, sup v∈S n-1 ∂ + v d ∂Ω (x)),
where ∂ + v d ∂Ω (x) denotes the directional derivative of d ∂Ω in the direction v. What is more, as it can be seen from [23, Section 2.1], we have the following formula for the λ-medial axis.

M λ (Ω) = x ∈ Ω : |∇d ∂Ω (x)| 2 ≤ 1 - λ 2 d 2 ∂Ω (x)
.

Motivated by this formula, we will prove the following.

Theorem b.5 (Theorem 1.3.1). Let Ω be a smooth bounded open subset of R n . For m > 0, let v m be the (unique) solution of the variational problem:

inf Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 .
Let us dene the set E m,λ by:

E m,λ := x ∈ Ω : |∇v m (x)| 2 ≤ 1 - λ 2 v 2 m (x) .
Then, for any λ > 0 and ε > 0, we have Theorem b.6 (Theorem 1.3.2). Let C d > 0 be such that d ∂Ω is C d -semiconcave. There exists m 0 > 0 such that for any m > m 0 , the solution v m of the elasto-plastic torsion problem (b.1) is C d -semiconcave.

The proof of Theorem 1.3.2 relies on the two following facts:

• inside E m , v m has constant Laplacian,

• outside E m , v m is equal to d ∂Ω , which is C d -semiconcave, for some constant C d > 0 that depends only on Ω.

The main tool we use is Korevaar's convexity maximum principle [START_REF] Korevaar | Convex solutions to nonlinear elliptic and parabolic boundary value problems[END_REF]Theorem 1.3]. Let us present this result. Let u : Ω → R be a C 2 function satisfying an equation of the form P (x, u(x), ∇u(x), D 2 u(x)) = 0

in Ω, where P is a certain elliptic operator, and D 2 u is the Hessian of u. The convexity function of u is the function c dened for any [x, y] ⊂ Ω and λ ∈ [0, 1] by c(x, y, λ) := (1 -λ)u(x) + λu(y) -u((1 -λ)x + λy).

Under suitable conditions on the elliptic operator P , Korevaar's convexity maximum principle states that the minimum of c is attained at a triple (x, y, λ) such that one of the points x, y or (1 -λ)x + λy belongs to the boundary of Ω. In particular, this result applies to the case where P is the euclidean Laplacian. Then, J is called a Jacobi eld if there exists a variation of γ through geodesics whose J is the variation eld, i.e. a smooth map Γ : (-1, 1) × [a, b] → M such that Γ(0, •) = γ and for any s ∈ (-1, 1), Γ(s, •) is a geodesic, and

∂ 1 Γ(0, t) = J(t).
Furthermore, if there exists a Jacobi eld J along γ such that J(a) = 0 and J(b) = 0, then we say that γ(a) and γ(b) are conjugate points along γ. Denition c.2. Let T > 0 and γ : [0, T ] → M be a unit speed geodesic such that γ(0) = b, t 0 ∈ (0, T ) and p = γ(t 0 ). We say that p is a cut point of b along γ if γ is length minimizing between b and p, but not after p, i.e d b (γ(t)) = t for t ≤ t 0 , and d b (γ(t)) < t for t > t 0 . A point q ∈ M is said to be a cut point of b if there exists a geodesic γ such that q is a cut point of b along γ. We recall the following. Denition c.4. Let q ∈ M and V ∈ T q M be such that there exists a well dened geodesic γ : [0, 1] → M such that γ(0) = q and γ(0) = V . Then, the (global) exponential map at V is dened by Exp(V ) = γ [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]. The restriction of the exponential map Exp on T q M is denoted by exp q and called the exponential map at q.

The following facts can all be found in [74, Chapter III, Section 4]: Proposition c.5. Some well known facts related to the cut locus include:

• Cut b (M ) is the set of points p in M , such that either there exist at least two minimizing geodesics from b to p, or the points b and p are conjugate,

• Cut b (M ) is the closure of the set of points p in M , such that there exist at least two minimizing geodesics from b to p,

• the function d b is smooth outside Cut b (M ) ∪ {b}, and it is dierentiable at a point p if and only if there exists a unique minimizing geodesics between b and p,

• in particular, Cut b (M ) ∪ {b} is the closure of the set of points of non-dierentiability of d b ,

• the exponential map exp b is a dieomorphism from an open set of

T b M onto M \ Cut b (M ),
• Cut b (M ) is a deformation retract of M \ {b}. In particular, these two sets have the same homotopy type, and so Cut b (M ) inherits much of the topology of M (like homology groups, for instance). See [74, Chapter III, Section 4, proposition 4.5] for a precise statement.

We know quite a bit about the structure of the cut locus. Let us give some of its properties, to get an idea of what the cut locus looks like. First, it is the union of smooth hypersurfaces consisting of points with exactly two minimizing geodesics, plus a set of Hausdor dimension at most n -2. A reader-friendly reference for this is [START_REF] Itoh | The dimension of a cut locus on a smooth Riemannian manifold[END_REF]. Furthermore, the cut locus has locally nite n -1-dimensional Hausdor measure, see for instance [START_REF] Itoh | The Lipschitz Continuity of the Distance Function to the Cut Locus[END_REF]Theorem B]. To our knowledge, the most precise results in this direction are contained in [START_REF] Ardoy | Cut and singular loci up to codimension 3[END_REF], where points of the cut locus are classied, up to a set of Hausdor dimension at most n -3. We refer to [START_REF] Ardoy | Cut and conjugate points of the exponential map, with applications[END_REF]Section 1.3] for additional structural properties of the cut locus.

An important property of the distance function is semiconcavity. Using some regularity result ( [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations in innite dimensions. IV. Hamiltonians with unbounded linear terms[END_REF]) for Hamilton-Jacobi equations, they deduced that for any local chart ψ of M , the map d b • ψ -1 is locally semiconcave as a function on R n (see also [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] for a more reader-friendly reference about semiconcave functions and Hamilton-Jacobi equations). In the Appendix A.1 about semiconcave functions, we show that it is equivalent to saying that d b is locally semiconcave. A dierent proof of the local semiconcavity of distance functions can also be found in [START_REF] Albano | Singular gradient ow of the distance function and homotopy equivalence[END_REF]. In [START_REF] Mantegazza | On the distributional Hessian of the distance function[END_REF] (and in particular Theorem Theorem c.8. The distributional Hessian of d b is given by:

D 2 d b = D 2 d b Vol -(ν ⊗ ν) ∇d + b -∇d - b H n-1 Cut b (M ) ,
where Denition c.9. Let ψ : M → R be a continuous function, and p ∈ M . We say that the Laplacian of ψ at p is -∞ in the sense of barriers if for any A > 0, there exists a smooth function φ : M → R dened on a neighborhood of p such that φ ≥ ψ, φ(p) = ψ(p) and ∆φ(p) ≤ -A.

We will prove the following. This result proves useful when one wants to apply the maximum principle to a function involving the distance function d b . It will be applied to the study of a variation of the elasticplastic torsion problem on manifolds, that we will now introduce.

d A variation of the elastic-plastic torsion problem on manifolds

Let us now assume that the manifold M is compact. The cut locus is a fundamental object in Riemannian geometry, and it is a natural problem to try and nd ways to compute it numerically. While the medial axis of Ω is unstable with respect to non-smooth variation of the boundary of Ω, the cut locus is unstable with respect to non-smooth variations of the metric of M . Indeed, it is proved in [START_REF] Albano | On the stability of the cut locus[END_REF] that the cut locus is unstable for some C 1 variations of the metric of M (but stable for C 2 variations of the metric of M ). In particular, the cut locus of a surface S cannot simply be approximated with the cut locus of a triangulation of S. Our strategy will be to extend the results presented above linking the λ-medial axis and the elasticplastic torsion problem to the case of the cut locus, and to derive a numerical approximation of the cut locus of triangulated surfaces. We will study the following variational problem in Chapter 3:

inf M |∇u| 2 -mu : u ∈ H 1 (M ), |∇u| ≤ 1, u(b) = 0 , (d.1)
where m > 0 is a given constant. We will extend the results for the elastic-plastic torsion problem presented earlier in this introduction.

Remark d.1. We could have replaced the point b with a smooth hypersurface S ⊂ M in the above problem. However, our interest here is in the instability of the cut locus with respect to the metric of M , and this aspect is already present with a point b instead of S. So for simplicity, we focus on the case of the cut locus of a point.

Note that by convexity of the functional, this problem has a unique minimizer u m . One may ask if an analog of Theorem b.1 is also true for u m , with d ∂Ω replaced with d b and Ω with M . Interestingly, it is not. Theorem d.2 (Theorem 3.3.1). There exists a surface of revolution M and a parameter m > 0 such that, if u d m denotes the unique minimizer of the following problem:

inf M |∇u| 2 -mu : u ∈ H 1 (M ), u ≤ d b ,
then we have u m = u d m .

However, we prove that the equality holds if m is large enough, depending on the Ricci curvature of M .

Theorem d.3 (Theorem 3.3.2). Let K ≥ 0 be such that

Ric ≥ -K,
where Ric denotes the Ricci curvature tensor of the manifold M . If

m ≥ max( nK(1 + Kdiam(M ) 2 ), nKdiam(M )),
then ∇u d m ≤ 1, or equivalently

u d m = u m .
The technique of the proof is dierent from the one in [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF], which was specic to the at case.

{|∇u m | < 1} -→ m→+∞ Cut b (M ).
Once we have the inclusion, the proof of the convergence is simple. It is based on the maximum principle, applied to the right function. Note to this end that u m veries on the whole manifold M , ∆u m ≥ -2m, in the sense of distributions. To prove the inclusion, we will use the maximum principle and the Theorem c.10 about the Laplacian of the distance function on the cut locus.

We will now state our analogous result to Theorem b.5 for problem (d.1). This result will be used in Chapter 4 to compute a stable approximation of the cut locus. We recall that the Following the analogous formula for the λ-medial axis, we dene a λ-cut locus by

Cut b (M ) λ := x ∈ M : |∇d b (x)| 2 ≤ 1 - λ 2 d 2 b (x)
, for λ > 0.

It is clear that the map λ → Cut b (M ) λ is non-increasing, and we have

Cut b (M ) = λ>0 Cut b (M ) λ .
Recall that u m denotes the minimizer of (d.1). We set

E m := x ∈ M : |∇u m (x)| < 1 , and E m,λ := x ∈ M : |∇u m (x)| 2 ≤ 1 - λ 2 u 2 m (x)
.

We will prove the following. Theorem d.5 (Theorem 3.1.2). There exists m 0 > 0, depending only on the manifold M , such that for any ε > 0, there exists a constant C > 0 such that, for any m > m 0 ,

u m is C-semiconcave on M \ B(b, ε).
What is more,

sup x∈E m,λ d(x, Cut b (M ) λ ) -→ m→+∞ 0, and sup x∈Cut b (M ) λ+ε d(x, E m,λ ) -→ m→+∞ 0.
Once the uniform semiconcavity is proved, the rest of the theorem is strictly similar to the corresponding one for the classical elastic-plastic torsion problem. The proof of semiconcavity, however, is dierent. Indeed, in this case, we cannot apply Korevaar's maximum principle to prove the semiconcavity in E m . It can be seen that proving the semiconcavity is equivalent to nding a constant C such that for any p ∈ E m and X ∈ T p M of norm 1, we have D 2 u m (X, X) ≤ C. Our strategy will be to assume that there exists p ∈ E m and X ∈ T p M that maximizes the quantity D 2 u m (X, X). Then, we extend X by a well chosen vector eld X, and nd a suitable subharmonic function involving D 2 u m ( X, X). This technique is inspired from the proof of second order estimate for hessian equations in Riemannian manifolds by Guan in [START_REF] Guan | Second Order Estimates and Regularity for Fully Nonlinear Elliptic Equations on Riemannian Manifolds[END_REF].

Numerical approximation. Now let us consider the case M = S, a compact real analytic surface without boundary embedded in R 3 . In Chapter 4, we will use the above results to compute a numerically stable approximation of the cut locus of S. A few authors have done it before, with dierent techniques. These techniques may be classied into two categories.

Geodesic approximation on parametrized surfaces. This approach was used in [START_REF] Sinclair | Loki: Software for computing cut loci[END_REF] and [START_REF] Misztal | Cut locus construction using deformable simplicial complexes[END_REF].

In [START_REF] Sinclair | Loki: Software for computing cut loci[END_REF], on genus 1 parametrized surfaces, the authors computed a degree 4 polynomial approximation of the exponential map using the geodesic equation, and deduced an approximation of the cut locus from there. In [START_REF] Misztal | Cut locus construction using deformable simplicial complexes[END_REF], the authors used the deformable simplicial complexes (DSC) method and nite dierences techniques for geodesic computations, to compute geodesic circles of increasing radius and their se-intersection, i.e. the cut locus. They apply the method to genus 1 surfaces. These papers contain no proof of convergence of the computed cut locus.

Exact geodesic computation on discretized surfaces. This approach was used in [START_REF] Itoh | Thaw: A tool for approximating cut loci on a triangulation of a surface[END_REF] and [START_REF] Dey | Cut locus and topology from surface point data[END_REF].

In [START_REF] Itoh | Thaw: A tool for approximating cut loci on a triangulation of a surface[END_REF], the authors computed the geodesics on a convex triangulated surface. They deduced an approximation of the cut locus of the triangulated surface, and ltered it according to the angle formed by the geodesics meeting at a point of the approximated cut locus, to make their approximation stable. They applied the method to ellipsoids. There is no proof of convergence.

In [START_REF] Dey | Cut locus and topology from surface point data[END_REF], the authors computed shortest curves on a graph obtained from a suciently dense sample of points of a surface. From there they deduced an approximation of the cut locus, and ltered it according to the maximal distance (called spread) between the geodesics meeting at a point of the approximated cut locus. They proved that the set they compute converges to the cut locus (see [START_REF] Dey | Cut locus and topology from surface point data[END_REF]Theorem 4.1]).

We also mention [START_REF] Bonnard | Geometric and numerical techniques to compute conjugate and cut loci on riemannian surfaces[END_REF], where the authors use some more geometric tools to compute (numerically) the cut locus of an ellipsoid, or a sphere with some particular metric with singularities.

The strategy we use is quite dierent. By the above results, for λ > 0 to be 'chosen small', and m 'large enough', the set E m,λ : Furthermore, we will show (see Proposition 4.2.2) that for λ small enough, Cut b (S) λ has a connected component that is homotopy equivalent to Cut b (S). On a triangulation S h of the surface S, the minimizer u m can be approximated with a function u h by the nite element methods. We will prove the following convergence in measure (see Proposition 4.4.4):

= |∇u m | 2 ≤ 1 -λ 2
|∇u h | 2 ≤ 1 - λ 2 u 2 h -→ h→0 E m,λ .
The eciency of the method is illustrated on complex geometries in Section 4.5.

e A non-local isoperimetric problem

In part II, our focus will be on a non-local isoperimetric problem.

The classical isoperimetric problem. Let n ∈ N * be an integer. For any measurable set E ⊂ R n , the perimeter of E is dened as the total variation of the indicator function of E:

P (E) := R n |Dχ E | = sup E div f : f ∈ C 1 c (R n ), |f | L ∞ ≤ 1 .
The quantity P (E) may be innite, and when it is nite, we say that E is a set of nite perimeter. If E is a C 1 open set, then as one would expect, we have P (E) = H n-1 (∂E), where H n-1 is the (n -1)-dimensional Hausdor measure. This denition is due to De Giorgi.

It allowed him, using Steiner symmetrization, to give a complete proof of the isoperimetric inequality. The classical isoperimetric problem is the following variational problem:

min E⊂R n ,|E|=m P (E),
where |E| is the volume of E, and m > 0 is a constant. It is now well known that for any m > 0, this problem has a unique minimizer up to translation, namely B[m], the ball of volume m. Moreover, a sharp quantitative estimate was proved in [START_REF] Fusco | The Sharp Quantitative Isoperimetric Inequality[END_REF] by Fusco, Maggi and

Pratelli. Before stating it, we need a denition. Given a set E of volume m, we call Fraenkel asymmetry the quantity λ(E) := min

x∈R n |E∆(x + B[m])| |E| , where E∆F = (E ∪ F ) \ (E ∩ F )
is the symmetric dierence of E and F . There exists a constant C(n) depending only on the dimension of the ambient space such that for any set E of volume m, P

E) -P (B[m]) ≥ C(n)P (B[m])λ(E) 2 . (
See also [START_REF] Fusco | A strong form of the quantitative isoperimetric inequality[END_REF] for a renement of this inequality, where the dierence P (E) -P (B[m]) also controls the oscillation of the normal to the (reduced) boundary of E. The quantitative isoperimetric inequality plays an instrumental role in the study of minimizers of small masses of Gamow's liquid drop model. Our book of reference for the perimeter is Giusti's book, [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF].

Maggi's book [START_REF] Maggi | Sets of nite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF] is probably more reader-friendly though (but does not include the study of BV functions).

Gamow's liquid drop problem. Gamow's liquid drop model for the atomic nucleus consists in:

inf

E⊂R 3 ,|E|=m P (E) + E×E dxdy |x -y| , where • P (E) is the perimeter of E,
• m, called the mass, is a positive constant.

In this model, the nucleus is thought of as an incompressible charged liquid. The perimeter term models the short-range attractive nuclear force, whereas the other term, similar to an electrostatic energy, models the repulsive forces between the protons of the nucleus. Gamow invented this model in the 1930's, in an attempt to predict the shape of nuclei, the nonexistence of large nuclei, and the existence of a nucleus with the least energy per nucleon. These three goals have only been attained relatively recently (in the 2010's, see below).

Some variations of the liquid drop model.

P (E) + V α (E) (e.1)
where

• P (E) is the perimeter of E, • V α (E) := E×E dxdy |x-y| n-α is the Riesz energy of E,
• n ≥ 2 (the dimension of the ambient space), m > 0 and α ∈ (0, n) are constants.

The physical case corresponds to n = 3 and α = 2. In this case, the Riesz energy boils down to the electrostatic energy E×E dxdy |x-y| .

It was shown in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] by Figalli, Fusco, Maggi, Millot, and Morini, that if the mass m is small enough, then the problem (e.1) admits a unique minimizer (up to translation), namely the ball of volume m. See also the papers of Knüpfer, Muratov, Julin, Bonacini and Cristoferi [START_REF] Knüpfer | On an isoperimetric problem with a competing nonlocal term i: The planar case[END_REF], [START_REF] Muratov | On an isoperimetric problem with a competing nonlocal term ii: The general case[END_REF], [START_REF] Julin | Isoperimetric problem with a Coulombic repulsive term[END_REF] and [START_REF] Bonacini | Local and global minimality results for a nonlocal isoperimetric problem on R N[END_REF], for anterior and slightly less general results.

Another interesting variation consists in replacing the repulsive Riesz energy with a repulsive energy involving another kernel. Let K ∈ L 1 (R n ). Problem (e.1) can be rephrased as inf E⊂R n ,|E|=m

P (E) + E×E K(x -y)dxdy,
with K(x) = |x| α-n . In [START_REF] Rigot | Ensembles quasi-minimaux avec contrainte de volume et rectiabilité uniforme. (Quasi-minimal sets with a volume constraint and uniform rectiability)[END_REF], Rigot considered the above problem with K an L 1 compactly supported function, and proved that it has a minimizer for any mass m. The following papers are posterior to the present work. In [START_REF] Pegon | Large mass minimizers for isoperimetric problems with integrable nonlocal potentials[END_REF], Pegon proved the same result in the case of radially symmetric non-increasing kernels K with some regularity and integrability conditions at innity. He also showed that, under the same conditions, after rescaling, the problem Γ-converges to the classical isoperimetric problem as the mass m goes to +∞. In [START_REF] Muratov | A Nonlocal Isoperimetric Problem with Dipolar Repulsion[END_REF], Muratov and Simon considered the kernel |x| -3 in dimension 2, and also studied the minimizers of large mass.

In [START_REF] Alama | Droplet breakup in the liquid drop model with background potential[END_REF], Alama, Bronsard, Choksi, and Topaloglu studied problem (e.1) with the addition of a conning background potential of the form -Z |x| p . They proved that it admits a minimizer for any mass m, and study the minimizers as Z → 0.

In [START_REF] Goldman | Quantitative estimates for bending energies and applications to non-local variational problems[END_REF], Goldman, Novaga and Roger studied the small mass minimizers of problem (e.1)

with the addition of the Willmore energy.

As shown in [START_REF] Choksi | Small volume-fraction limit of the diblock copolymer problem: Ii. diuse-interface functional[END_REF], Gamow's liquid drop model is also related to diblock copolymers, and the OhtaKawasaki functional. This is also an active eld of research, see for instance [START_REF] Alama | Ground-states for the liquid drop and tfdw models with long-range attraction[END_REF] and [START_REF] Van Den Berg | Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem[END_REF].

Our problem. To restore the existence of a minimizer for large masses in (e.1), we will add the energy associated to the potential A |x| β to our functional, as we expect it to counter the spreading eect of the V α term. Thus we will be interested in the following modication of the original problem (e.1):

inf

E⊂R n ,|E|=m E α,β,A (E) := P (E) + V α (E) + U β,A (E), (e.2)
where

• U β,A (E) := E A |x| β dx,
• A ≥ 0 and β > 0 are constants.

Indeed, an elementary proof will show that this extra connement potential (no pun intended) restores the compactness of a minimizing sequence for any mass m. Therefore, this problem always has a minimizer. We will then be interested in the behavior of these minimizers when the mass m goes to +∞, with particular emphasis in the case β > α. We will prove the following.

Theorem e.1 (Theorem 1.1.1). Given α ∈ (0, n), β > 0 and A > 0, assume α < β. Let (E m ) m>0 be a family of minimizers in (1.1.3), such that |E m | = m, and let E * m be the rescaling of E m of the same mass as the unit ball B centered at 0 (ie

E * m = |B| m 1 n E m ).
Then, up to modifying the sets (E m ) by some sets of measure 0, the boundaries of the sets (E * m ) Hausdorconverge to the boundary of B as m → +∞.

Using the techniques of [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF], one can see that in this case, if β > 1 then balls of large masses are locally minimizing. On the contrary if β < 1 then balls of large masses are not locally minimizing. By locally minimizing, we mean that a set E is a local minimizer if a small L 1 perturbation of E always has greater energy than E. With the additional assumption that β > 1 we will actually prove that minimizers of large masses are exactly balls. Theorem e.2 (Theorem 1.1.2). Given α ∈ (0, n), β > 0 and A > 0, assume 1 < α < β. There exists a mass m 1 = m 1 (n, α, β, A) > 0 such that for any m > m 1 the ball of volume m centered at 0 is the unique minimizer (1.1.3), up to a set of measure 0.

We conjecture that this last theorem holds also for any α ∈ (0, 1] as long as β > 1.

However, this seems to require dierent techniques.

We started working on some aspects of this functional under the supervision of Vincent Millot for our Master 2 thesis. In part II, the new results start with Section 1.4. In particular, the results presented in this introduction are new.

Numerical minimization In Chapter 2, we will develop a method to compute numerically the minimizers of (e.2). In particular, we will apply this method to the original problem (e.1).

Indeed, the theoretical knowledge we have so far on problem (e.1) raises two natural questions.

Is it always the case (i.e. for any value of α ∈ (0, n)) that there is no minimizer for m large enough? Is there a set of parameters n, α and m, such that there exists a minimizer that is dierent from a ball? Our numerical results indicate that in dimension 2, the answers are positive and negative respectively, as conjectured by Choksi and Peletier in [START_REF] Choksi | Small volume-fraction limit of the diblock copolymer problem: Ii. diuse-interface functional[END_REF]Conjecture 6.1] Γ-convergence. To justify our numerical method, we will prove some Γ-convergence results. This is the notion of convergence of functionals one generally wants to use when studying variational problems. Denition e.3. Let X be a topological space, F a real functional on X, and (F ε ) ε>0 a family of real functionals on X. We say that F ε Γ-converges to F as ε goes to 0, and we write

F ε Γ ---→ ε→0
F, if the two following assertions hold:

• for any u ∈ X and any family (u ε ) ε>0 of X such that u ε -→ ε→0 u, we have

lim inf ε→0 F ε (u ε ) ≥ F(u),
• for any u ∈ X, there exists a family (u ε ) ε>0 of X such that u ε -→ ε→0 u and

lim ε→0 F ε (u ε ) = F(u).
Remark e.4. We also say that a sequence of functionals (F N ) Γ-converges to a function F if the assertions above hold with ε and 0 replaced with N and +∞.

This denition guarantees that if, for any ε > 0, u ε is a minimizer of F ε in X, and the family (u ε ) converges to u ∈ X as ε → 0, then u is also a minimizer of F in X. To minimize numerically a given functional F, the general strategy is the following:

1. nd a sequence of discrete functionals (F N ) N ≥0 that Γ-converges to F, 2. nd numerically a minimizer u N of F N for some N considered 'large', 3. extrapolate a limit point u of the sequence (u N ) N ≥0 from the results. This u should be a minimizer of F.

For point 3 to even have a chance to work, we need the sequence (u N ) N ≥0 to be precompact:

Terminology e.5. Following [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF], we say that a family of functionals (F ε ) ε>0 dened on a metric space X enjoys property (C) (for compactness) if for any family (u ε ) ε>0 of elements of X such that (F ε (u ε )) ε>0 is bounded, there is a subsequence of (u ε ) ε>0 that converges in X.

If a family of functionals (F ε ) ε>0 enjoys property (C) and Γ-converges to a limit functional F when ε goes to 0, then we know that for ε small enough, minimizers of F ε are close to minimizers of F. When discretizing our functional F, along with the Γ-convergence, we will prove that the property (C) is veried. We refer to the book [START_REF] Braides | Gamma-convergence for Beginners[END_REF] for more details about Γ-convergence. The lecture [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF] also contains a brief introduction to Γ-convergence that is sucient to our needs.

Modica-Mortola theorem. The numerical treatment of the perimeter is classical. To deal with the perimeter term, we will use the well known Modica-Mortola theorem. This theorem allows us to relax the perimeter functional on sets, i.e. charateristic functions, into a functional on functions taking values in [0, 1]. Therefore we will be able to use the vector space structure of functions and, after disctretization (step three below), usual optimization tools for functionals on R d . Theorem e.6 (Modica-Mortola theorem.). Let Ω be a Lipschitz open bounded subset of R n and ε > 0. Let W be the double well potential dened on [0, 1] by W (x) = x(1 -x). Let us set σ := 2 1 0 W (u)du. We dene the set X, and the functionals F ε : X → R by

X := u ∈ L 1 (Ω, [0, 1]) : u = m ,
endowed with the strong L 1 -topology, and

F ε (u) = σ -1 R n ε |∇u| 2 + ε -1 W (u) if u ∈ H 1 0 (Ω, [0, 1]), +∞ otherwise, (e.3) If (u ε ) ε>0 is a family of functions in H 1 0 (Ω, [0, 1]) and E ⊂ Ω is any measurable subset of Ω, such that u ε -→ ε→0 1 E in L 1 (Ω), then we have P (E) ≤ lim inf ε→0 F ε (u ε ).
Moreover, if E ⊂⊂ Ω is a compact subset of Ω of nite perimeter, then there exists a family of functions (u ε ) ε>0 in X, such that u ε -→ ε→0 1 E , and

P (E) = lim ε→0 F ε (u ε ).
Finally, the family of functionals (F ε ) enjoys property (C).

One can nd a reader-friendly proof of these facts in [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF]. Here, we will only give the following heuristic. If ε > 0 is very small, then ε -1 W (u) is very large, except if u is 0 or 1. Therefore the minimizers of F ε are functions that are equal to 0 or 1 except on a set of small measure.

Method of the numerical minimization. Here are the modications of the variational problem (1.1.3) that we will use in Chapter 2 to arrive at a nite dimensional variational problem that can be easily numerically solved. All steps will be justied by a Γ-convergence and compactness result.

Step one is standard when dealing with the perimeter. We use the classical Modica-Mortola theorem to relax the functional on sets, i.e. charateristic functions, into a functional on functions taking values in [0,[START_REF] Adelstein | Morse theory for the uniform energy[END_REF]. This allows us to use the vector space structure of functions and, after disctretization (step three), usual optimization tools for functionals on R d .

Step two is the key step for dealing with the non-local term V α . We will replace the ambient space R n with a large square with periodic boundary conditions, whose size is a new relaxation parameter. Then, we can approximate the non-local term V α by a simple expression in Fourier variable.

In step three, we discretize the problem by considering only trigonometric functions with frequencies lower than some integer N , and by computing the integral terms with riemann sums.

The numerical results are presented in section 2.3.

Part I

Elastic-plastic torsion problem, distance function and cut locus

Outline of part I

This part is independent of part II. We will try and give a global view of the content of this part. Then, each Chapter will start with a more detailed specic introduction, giving also more context about their respective topics.

We are interested in the approximation of the cut locus of a Riemannian manifold through a variational problem. Given a smooth Riemannian manifold M and a point b in M , the cut locus of b in M , denoted by Cut b (M ), is the set of points at which geodesics starting from b stop being minimizing. Alternatively, it is also the set of points around which the distance function to b, denoted by d b , is not smooth. Or the closure of the set of points p such that there exist at least two minimizing geodesics from b to p. Alternatively, it is the set of points at which the distance function to ∂Ω is not dierentiable.

The elastic-plastic torsion problem is the following variational problem:

inf

Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 . (0.1)
It has been intensively studied in the 60's and 70's by various mathematicians. See Section 1.1 for more details. Let v m be its unique minimizer. In [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF], in particular, it was proved that the set {|∇v m | < 1} contains the set M(Ω), and that the former converges in the Hausdor sense to the latter as m goes to +∞. Like the cut locus, the medial axis has some kind of instability. Indeed, M(Ω) is unstable with respect to non-smooth variations of the boundary of Ω (see Figure 1 what they did in their paper [START_REF] Chazal | The λ-medial axis[END_REF]. See [START_REF] Saha | Chapter 1 -Skeletonization and its applications a review[END_REF] for many more ways of computing the medial axis, and applications. Another useful formula for M λ (Ω) can be derived using the generalized gradient of d ∂Ω . Indeed, d ∂Ω is locally semiconcave. This means that, locally, there exists a constant C > 0 such that the function x → C |x| 2 -d ∂Ω (x) is convex. In particular, as convex functions, it has a well dened generalized gradient. See Appendix A.1 for more details. Now, one can see that the λ-medial axis is also given by the formula

M λ (Ω) = x ∈ Ω : |∇d ∂Ω (x)| 2 ≤ 1 - λ 2 d 2 ∂Ω (x)
.

Motivated by these facts, we prove that the set E m,λ dened by

E m,λ := x ∈ Ω : |∇v m (x)| 2 ≤ 1 - λ 2 v 2 m (x)
is basically a good approximation of M λ (Ω) if m is large enough.

Theorem (Theorem 1.3.1). Let v m be the unique solution of the elastic-plastic torsion problem (0.1). Then, for any λ > 0 and ε > 0, we have

sup x∈E m,λ d(x, M λ (Ω)) -→ m→+∞ 0 and sup x∈M λ+ε (Ω) d(x, E m,λ ) -→ m→+∞ 0.
This means that a stable version of the cut locus can be approximated from the minimizer of a convex variational problem.

The strategy of the proof is to show that the minimizers (v m ) m>0 of the elastic-plastic torsion problem are uniformly semiconcave when m is large enough.

Theorem (Theorem 1.3.2). Let C d > 0 be such that d ∂Ω is C d -semiconcave. There exists m 0 > 0 such that for any m > m 0 , the solution v m of the elasto-plastic torsion problem (0.1) is C d -semiconcave. This is done by using Korevaar's convexity maximum principle [START_REF] Korevaar | Convex solutions to nonlinear elliptic and parabolic boundary value problems[END_REF].

The idea is then to apply the same strategy to the study of the cut locus and the problem of instability with respect to the metric, and to derive a method for computing the cut locus of a surface. In Chapter 3, we study the following variational problem:

inf M |∇u| 2 -mu : u ∈ H 1 (M ), |∇u| ≤ 1, u(b) = 0 . (0.2)
It is a kind of generalization of the elastic-torsion problem, where the ambient euclidean space is replaced with a smooth compact Riemannian manifold without boundary M , and the boundary of ∂Ω is replaced with the point b. In this framework, the analog of M(Ω) is Cut b (M ). We extend some results known for the elastic-plastic torsion problem to this framework. Let Cut b (M ) λ ⊂ M be dened by

Cut b (M ) λ := p ∈ M : |∇d b (x)| 2 ≤ 1 - λ 2 d 2 b (x)
, where ∇d b (x) is the generalized gradient of d b (see Propositions A.1.9 and A.1.6). We will prove the following theorem.

Theorem (Theorem 3.1.2). There exists m 0 > 0, depending only on the manifold M , such that for any m > m > m 0 , the minimizer u m of (0.2) is locally C 1,1 on M \ {b}, and

Cut b (M ) ⊂ {|∇u m | < 1} ⊂ {|∇u m | < 1}.
Moreover, M ) in the Hausdor sense.

E m -→ m→+∞ Cut b (
Given any ε > 0, there exists a constant C > 0 such that, for any m > m 0 ,

u m is C-semiconcave on M \ B(b, ε). Finally, sup x∈E m,λ d(x, Cut b (M ) λ ) -→ m→+∞ 0, and sup x∈Cut b (M ) λ+ε d(x, E m,λ ) -→ m→+∞ 0.
This result will be used in chapter 4 to build a new method for the numerical approximation of the cut locus of a surface.

In our study, we will need to establish that the gradient constraint |∇u| ≤ 1 in (0.2) can be replaced with the obstacle constraint u ≤ d b . Contrary to the case of the classical obstacle problem (0.1), this is not true in general. However, it is true if m is large enough. Indeed, let u d m be the minimizer of the following variational problem:

inf M |∇u| 2 -mu : u ∈ H 1 (M ), u ≤ d b .
We will prove the following.

Theorem (Theorem 3.3.1). There exists a surface of revolution M and a parameter m > 0 such that u m = u d m . Theorem (Theorem 3.3.2). Let K ≥ 0 be such that

Ric ≥ -K, (0.3)
where Ric denotes the Ricci curvature tensor of the manifold M . If

m ≥ max( nK(1 + Kdiam(M ) 2 ), nKdiam(M )), then u d m = u m .
In Chapter 4, we address the question of the numerical approximation of the cut locus based on previous modeling. We introduce a nite element method based on triangulated surfaces, prove the convergence of our scheme and illustrate its eciency on complex geometries.

In Chapter 2, after introducing the notion of cut locus, and some important properties of Remark 0.1. We could have replaced the point b with a smooth hypersurface S ⊂ M in all our considerations. However, our interest here is in the instability of the cut locus with respect to the metric of M , and this aspect is already present with a point b instead of S. So for simplicity, we focus on the case of the cut locus of a point. The elastic-plastic torsion problem.

In this chapter, Ω is a smooth bounded open set of R n . The elastic-plastic torsion problem is the following variational problem, dened for any m > 0.

inf

Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 . (1.1.1)
One can show that this functional has a minimizer by the direct method of calculus of variations. The minimizer is unique because we are minimizing a strictly convex functional over a convex subset of H 1 0 (Ω). Let v m be the unique minimizer of (1.1.1). (See also [START_REF] Hartman | On some non-linear elliptic dierential-functional equations[END_REF] for general existence and uniqueness results for more general functionals of that type.) Physically speaking, v m represents the stress function of a long bar of cross section Ω, twisted with an angle m. The elastic-plastic torsion problem and the properties of its minimizer v m have been intensively studied in the 60s and 70s by various authors (such as Ting, Brézis, Sibony, Gerhardt, Caarelli, Rivière, Friedman, Pozzi...), see for instance [START_REF] Ting | Elastic-plastic torsion problem III[END_REF], [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF], [START_REF] Brezis | Multiplicateur de lagrange en torsion elasto-plastique[END_REF], [START_REF] Caarelli | The Smoothness of the Elastic-Plastic Free Boundary of a Twisted Bar[END_REF], [START_REF] Ting | Elastic-plastic torsion problem over multiply connected domains[END_REF], [START_REF] Caarelli | The lipschitz character of the stress tensor, when twisting an elastic plastic bar[END_REF], [START_REF] Friedman | The free boundary for elastic-plastic torsion problems[END_REF] and [START_REF] Caarelli | Unloading in the elastic-plastic torsion problem[END_REF].

In particular, we know from [START_REF] Caarelli | The lipschitz character of the stress tensor, when twisting an elastic plastic bar[END_REF] that v m is locally C 1,1 in Ω. What is more, we know from [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF] that the gradient constraint in (1.1.1) can be replaced with an obstacle type constraint involving the distance function d ∂Ω to the boundary of Ω. Indeed, we have

0 ≤ v m ≤ d ∂Ω and v m = arg min Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), v ≤ d ∂Ω . (1.1.2)
where

d ∂Ω (x) = min y∈∂Ω |x -y| .
Following [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF], we dene the elastic and plastic sets as follows:

E m = {x ∈ Ω : |∇v m (x)| < 1}, P m = {x ∈ Ω : |∇v m (x)| = 1}, 1.2. M λ (Ω): the λ-medial axis
It is then easy to prove (see [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF][START_REF] Alama | Droplet breakup in the liquid drop model with background potential[END_REF]) that

E m = {x ∈ Ω : v m (x) < d ∂Ω (x)}, P m = {x ∈ Ω : v m (x) = d ∂Ω (x)}. (1.1.3)
In the open set E m , we have the Euler-Lagrange equation of the variational problem:

∆v m = -2m in E m . (1.1.4) At a point x ∈ P m , we have v m (x) = d ∂Ω (x) and v m ≤ d ∂Ω around x, so ∇v m (x) = ∇d ∂Ω (x).
(1.1.5)

The medial axis of Ω, denoted by M(Ω), is dened as the set of points of Ω that have at least two closest points on the boundary ∂Ω of Ω:

M(Ω) := x ∈ Ω : ∃y, z ∈ ∂Ω, y = z and d ∂Ω (x) = |x -y| = |x -z| .
Equivalently, M(Ω) is the set of points of Ω at which the distance function d ∂Ω is not dierentiable. In [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF], Caarelli and Friedman used the notion of ridge of Ω, which is precisely the closure of M(Ω). Equivalently, it is the set of points around which the distance function is not smooth.

They proved the following.

Theorem 1.1.1 (Theorems 3.2 and 4.1 in [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF].). For any m > 0, we have

M(Ω) ⊂ E m .
Furthermore, there exists a constant C > 0 depending on Ω only such that

∀m > 0, d H (M(Ω), E m ) ≤ C m ,
where d H denotes the Hausdor distance.

We will also use the following fact. Equation (4.2) in [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF] gives a constant C > 0 depending only on

Ω such that ∀x ∈ Ω, d(x, {v m = d ∂Ω }) ≤ C m .
As d ∂Ω and v m are both 1-Lipschitz, we get that for a (dierent) constant C > 0, we have

∀m > 0, d ∂Ω -v m L ∞ (Ω) ≤ C m .
(1.1.6)

1.2

M λ (Ω): the λ-medial axis

The medial axis, also called skeleton, is used in a wide variety of applications in image processing and computer vision. Many researchers have proposed more or less ecient methods to compute it, and are still trying to improve its computation. We refer to the book [START_REF] Saha | Chapter 1 -Skeletonization and its applications a review[END_REF] from 2017 for a review about the medial axis, its computation and applications. In this section,

• Figure 1.1: A circle with its medial axis, and a polygonal approximation of a circle, with its medial axis.

we introduce the notion of λ-medial axis that was invented in 2005 by Chazal and Lieutier to give a theoretical framework to tackle the problem of stability.

Indeed, one important geometric property of the medial axis M(Ω) is that it is unstable with respect to small perturbations of the boundary of Ω -see Figure 1.1. This instability makes computing numerically M(Ω) quite tricky. Indeed, any numerical approximation of Ω (for instance, with polygons) might introduce an articial (and large) medial set. In order to deal with this problem, in [START_REF] Chazal | The λ-medial axis[END_REF], Chazal and Lieutier dened the so called λ-medial axis of Ω by setting, for any λ > 0,

M λ (Ω) := {x ∈ Ω : r(x) ≥ λ}, (1.2.1)
where r(x) is the radius of the smallest ball containing the set of all closest points to x on ∂Ω, i.e. the set {z ∈ ∂Ω :

|x -z| = d ∂Ω (x)}. Note that the map λ → M λ (Ω) is non-increasing,
and we have

M(Ω) = λ>0 M λ (Ω).
It is further proved in [23, Section 3, Theorem 2] that M λ (Ω) has the same homotopy type as M(Ω), for λ small enough. These facts justify that, for xed Ω, M λ (Ω) is a good approximation of M(Ω), for λ small enough. The crucial dierence though is that M λ (Ω) is stable with respect to small variations of Ω, whereas M(Ω) is not. Indeed we have the following theorem.

Theorem 1.2.1 (Consequence of Theorem 3 in [START_REF] Chazal | The λ-medial axis[END_REF]). For any 0 < λ < λ, there exists ε > 0 such that, if Ω and

Ω are two open bounded subset of R n such that d H (Ω c , Ω c ) < ε, then sup x∈M λ ( Ω) d(x, M λ (Ω)) ≤ 2 diam(Ω)ε,
where diam(Ω) is the diameter of Ω.

This means that although it is dicult to compute numerically M(Ω) directly, one can compute M λ (Ω) instead. In the same paper [START_REF] Chazal | The λ-medial axis[END_REF], the authors use voronoi diagrams to compute an approximation of M λ (Ω) from a noisy sample of points of the boundary of Ω.

We will need the following denition.

Denition 1.2.2. A function

u : Ω → R is said to be C-semiconcave (or just semiconcave) if the function x → C |x| 2 -u(x) is convex. This is equivalent to having, for any λ ∈ [0, 1] and x, y ∈ Ω such that [x, y] ∈ Ω, (1 -λ)u(x) + λu(y) -u((1 -λ)x + λy) ≤ Cλ(1 -λ) |y -x| 2 Then, we have: Lemma 1.2.3. Let Ω be a C 2 open bounded subset of R n . There is a constant C d > 0 such that d ∂Ω is C d -semiconcave on Ω.
This lemma is proved in [22, (iii) of Proposition 2.2.2], under slightly more general assumptions.

Here we only give the idea of the proof.

Idea of the proof. As Ω is of class C 2 , you can write the complement of Ω as a union of balls of radius r > 0 for some xed r. Then, d ∂Ω is the inmum of the distance functions to these balls of radius r. One can verify that the distance function to a ball is C-semiconcave for some C, and the supremum of a family of convex functions is convex, so d ∂Ω is also C-semiconcave.

As explained with more details in the Appendix A.1, this implies that, as concave functions, d ∂Ω has a well dened generalized gradient whose norm is given by the formula:

|∇d ∂Ω | (x) = max(0, sup v∈S n-1 ∂ + v d ∂Ω (x)), where ∂ + v d ∂Ω (x) denotes the directional derivative of d ∂Ω in the direction v. What is more, as it can be seen from [23, Section 2.1], we have |∇d ∂Ω (x)| 2 = 1 - r(x) 2 d ∂Ω (x) 2 ,
and so the following formula for the λ-medial axis holds.

M λ (Ω) = x ∈ Ω : |∇d ∂Ω (x)| 2 ≤ 1 - λ 2 d 2 ∂Ω (x)
.

(1.2.2)

In the next section, we will use this formula to approximate M(Ω) with the minimizer of the elastic-torsion problem presented in Section 1.1.

Elastic-plastic torsion problem and λ-medial axis

In this section, motivated by the two preceding sections, we prove the following renement of Theorem 1.1.1. It is the main result of this chapter.

Theorem 1.3.1. Let Ω be a smooth bounded open subset of R n . For m > 0, let v m be the unique solution of the variational problem:

inf Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 . (1.3.1)
Let us dene the set E m,λ by:

E m,λ := x ∈ Ω : |∇v m (x)| 2 ≤ 1 - λ 2 v 2 m (x) . (1.3.2)
Then, for any λ > 0 and ε > 0, we have

sup x∈E m,λ d(x, M λ (Ω)) -→ m→+∞ 0 and sup x∈M λ+ε (Ω) d(x, E m,λ ) -→ m→+∞ 0.
This theorem basically shows that the set E m,λ is a good approximation of the λ-medial axis.

In particular, it yields a new method for the numerical computation of the medial axis, based on the solution of a convex variational problem. In Chapter 3, we will use this strategy to approximate the cut locus of a manifold. The numerical results will be presented in Chapter 4.

The most important step in the proof of Theorem 1.3.1, is to show that the functions

(v m ) m>0 are uniformly semiconcave. Theorem 1.3.2. Let C d > 0 be such that d ∂Ω is C d -semiconcave. There exists m 0 > 0 such that for any m > m 0 , the solution v m of the elasto-plastic torsion problem (1.1.1) is C d -semiconcave.
With this uniform semiconcavity result, Theorem 1.3.1 follows from the uniform convergence of v m to d ∂Ω when m → +∞. The proof of 1.3.1 is almost identical to the proof of Proposition 3.5.6. We still give it here in full details.

The proof of Theorem 1.3.2 relies on the following two facts:

• inside E m , v m has constant Laplacian, • outside E m , v m is equal to d ∂Ω , which is C d -semiconcave, for some constant C d > 0 the depends only on Ω.
:= (1 -λ)f (x) + λf (y) -f ((1 -λ)x + λy).
Under suitable conditions on the elliptic operator P , Korevaar's convexity maximum principle states that the minimum of c is attained at a triple (x, y, λ) such that one of the points x, y or (1 -λ)x + λy belongs to the boundary of Ω. In particular, this result applies to the case where P is the euclidean Laplacian.

We are ready to prove Theorem 1.3.2. The rst part of the proof is identical to the rst part of the proof of Proposition 3.5.5. We still give it in full detail, for the reader's convenience.

Proof of Theorem 1.

3.2. Let C d be such that the distance function d ∂Ω is C d -semiconcave.
As Ω is smooth, every point of Ω in a neighborhood of ∂Ω has a unique projection on ∂Ω.

Therefore the distance between M(Ω) and ∂Ω is positive. From Theorem 1.1.1, we know that the elastic set E m converges to M(Ω) in the Hausdor sense when m goes to +∞. So there exists m 0 such that, for any m > m 0 , E m ⊂ Ω. Let ε > 0. We will show that for any m > m 0 , v m is (C d + ε)-semiconcave, which is sucient to conclude the proof. Let us introduce some notations. For a, b ∈ R and λ ∈ (0, 1), we set

λ ab := (1 -λ)a + λb. z f (z) • • • • x µ xy λ xy = ν µxyy y Figure 1
.2: Construction of γ and ν.

For any x, y ∈ Ω such that [x, y] ∈ Ω, λ ∈ [0, 1] and v a function on Ω, let us dene f v (z) := (C d + ε) |z| 2 -v(z), c(x, y, λ, v) := λ fv(x)fv(y) -f v (λ xy ). (1.3.3)
In order to prove that f vm is convex (i.e that v m is (C d + ε) semiconcave), we need to show the following:

inf x,y,λ c(x, y, λ, v m ) ≥ 0 (1.3.4)
where the inmum is taken over x, y ∈ Ω such that [x, y] ⊂ Ω and λ ∈ [0, 1]. Let us argue by contradiction and assume that (1.3.4) does not hold. By continuity of c, the inmum in (1.3.4) is a minimum, attained at a triple (x, y, λ), that veries c(x, y, λ, v m ) < 0.

Now we show that (x, y) ⊂ E m = {v m < d ∂Ω } . (1.3.5)
If not, then there exists µ ∈ (0, 1)

such that µ xy / ∈ E m , i.e v m (µ xy ) = d ∂Ω (µ xy ). Recalling that v m ≤ d ∂Ω , we get c(x, y, µ, v m ) ≥ c(x, y, µ, d ∂Ω ) > 0, (1.3.6)
where the last inequality comes from the C d -semiconcavity of d ∂Ω . In particular, µ = λ. Let us assume for instance that µ < λ (the case µ > λ being similar) and show that c(x, y, λ, v m )

is not minimal in (1.3.4). Figure 1.2 may help justify intuitively the following construction.

Let ν ∈ (0, 1) be such that ν µxyy = λ xy .

(1.3.7)

We have

c(µ xy , y, ν, v m ) = (1 -ν)f vm (µ xy ) + νf vm (y) -f vm (ν µxyy ) = (1 -ν)f vm (µ xy ) + νf vm (y) -f vm (λ xy ) = c(x, y, λ, v m ) -(1 -λ)f vm (x) + (ν -λ)f vm (y) + (1 -ν)f vm (µ xy ). (1.3.8)
Now after some elementary calculations, (1.3.7) translates into

1 -λ = (1 -ν)(1 -µ), ν -λ = -(1 -ν)µ, so (1.3.8) becomes c(µ xy , y, ν, v m ) = c(x, y, λ, v m ) -(1 -ν)((1 -µ)f vm (x) + µf vm (y) -f vm (µ xy )) = c(x, y, λ, v m ) -(1 -ν)c(x, y, µ, v m ) < c(x, y, λ, v m ), (1.3.9)
where in the last inequality we used the fact that c(x, y, µ, v m ) > 0 (equation (1.3.6)). This contradicts the minimality of c(x, y, λ, v m ). Thus (1.3.5) is true, and we are left to show that

v m is (C d + ε)-semiconcave inside E m .
The function

f vm : z → (C d + ε) |z| 2 -v m (z)
has constant Laplacian on E m . So we may apply Korevaar's convexity maximum principle (see [START_REF] Korevaar | Convex solutions to nonlinear elliptic and parabolic boundary value problems[END_REF]Theorem 1.3]) to get that the minimum of the function c(•,

•, •, v m ) on E m × E m × [0, 1]
is attained at a triple (x, y, λ) such that x, y, or λ xy is in ∂E m . As proved above, λ xy ∈ ∂E m is not an option, so x ∈ ∂E m or y ∈ ∂E m . Let us treat the case x ∈ ∂E m , for instance. We have chosen m 0 so that for m > m 0 , E m ⊂ Ω. In particular x ∈ Ω. We get that x ∈ P m , and in particular (recalling (1.1.3) and (1.

1.5)) v m (x) = d ∂Ω (x) and ∇v m (x) = d ∂Ω (x). (1.3.10) As d ∂Ω is C d -semiconcave, we have for any µ ∈ (0, 1), (1 -µ)(C d |x| 2 -d ∂Ω (x)) + µ(C d |y| 2 -d ∂Ω (y)) ≥ C d |(1 -µ)x + µy| 2 -d ∂Ω ((1 -µ)x + µy).
With (1.3.10), it yields as µ → 0,

(1 -µ)(C d |x| 2 -d ∂Ω (x)) + µ(C d |y| 2 -d ∂Ω (y)) ≥ C d |(1 -µ)x + µy| 2 -v m ((1 -µ)x + µy) + o(µ). Since v m ≤ d ∂Ω , (1 -µ)(C d |x| 2 -v m (x)) + µ(C d |y| 2 -v m (y)) ≥ C d |(1 -µ)x + µy| 2 -v m ((1 -µ)x + µy) + o(µ). (1.3.11)
Multiplying the following equality by ε

(1 -µ) |x| 2 + µ |y| 2 = |(1 -µ)x + µy| 2 + µ(1 -µ) |x -y| 2 ,
and adding it to (1.3.11), we get

(1 -µ)((C d + ε) |x| 2 -v m (x)) + µ((C d + ε) |y| 2 -v m (y)) ≥ (C d + ε) |(1 -µ)x + µy| 2 -v m ((1 -µ)x + µy) + εµ(1 -µ) |x -y| 2 + o(µ),
and so for µ small enough,

(1 -µ)((C d + ε) |x| 2 -v m (x)) + µ((C d + ε) |y| 2 -v m (y)) > (C d + ε) |(1 -µ)x + µy| 2 -v m ((1 -µ)x + µy),
i.e. c(x, y, µ, v m ) > 0. As before, we show that is implies

c(µ xy , y, ν, v m ) < c(x, y, λ, v m ),
where ν is such that ν µxyy = λ xy . This contradicts the minimality of c(x, y, λ). Therefore hypothesis (1.3.4) holds, which concludes the proof.

We can now prove Theorem 1.3.1.

Proof of Theorem 1.3.1. Let us argue by contradiction, and assume that the following convergence does not hold.

sup

x∈E m,λ d(x, M λ (Ω)) -→ m→+∞ 0 (1.3.12)
Let (m k ) k≥0 and (x k ) k≥0 be sequences in (0, ∞) and Ω respectively, such that m k -→ k→+∞ +∞ and for some η > 0, for any k ≥ 0,

x k ∈ E m k ,λ and d(x k , M λ (Ω)) > η.
(1.3.13)

Up to taking subsequences, we may assume that

(x k ) converges to a point x ∞ ∈ Ω. As x k ∈ E m k ,λ
lim inf k→∞ |∇v m k (x k )| ≥ |∇d ∂Ω (x ∞ )| .
Using the facts that x k ∈ E m k ,λ and v m k converges uniformly to d ∂Ω , this implies

|∇d ∂Ω (x ∞ )| 2 ≤ 1 - λ 2 d ∂Ω (x ∞ ) 2 ,
and so x ∞ ∈ M λ (Ω). But this contradicts (1.3.13). Thus (1.3.12) is proved.

We are left to prove the following onvergence:

sup x∈M λ+ε (Ω) d(x, E m,λ ) -→ m→+∞ 0. (1.3.14)
Once again, we argue by contradiction and assume that it does not hold. Let (m k ) k≥0 and (x k ) k≥0 be sequences of (0, ∞) and Ω respectively, such that m k -→ k→+∞ +∞ and for some η > 0,

for any k ≥ 0, x k ∈ M λ+ε (Ω) and d(x k , E m k ,λ ) > 2η. (1.3.15)
Up to taking subsequences, we can assume that (x k ) converges to a point x ∞ in M λ+ε (Ω). The last inequality implies that for any k large enough, we have

d(x ∞ , E m k ,λ ) > η.
According to (1.1.6), there exists a constant C 1 > 0 such that for any y ∈ Ω and k ≥ 0,

v m k (y) 2 ≥ d ∂Ω (y) 2 -C 1 /m k . As d ∂Ω is 1-Lipschitz and bounded, we deduce that there exists a constant C 2 > 0 such that for any y ∈ B(x ∞ , η) and k ≥ 0, v m k (y) 2 ≥ d ∂Ω (x ∞ ) 2 -C 2 η - C 1 /m k . Let η 0 > 0, k 0 > 0 and δ 0 > 0 be constants such that for any η ≤ η 0 and k ≥ k 0 , δ := (λ + ε) 2 d ∂Ω (x ∞ ) 2 - λ 2 d ∂Ω (x ∞ ) 2 -C 2 η -C 1 /m k ≥ δ 0 > 0.
For any y ∈ B(x ∞ , η), we have

1 - λ 2 v m k (y) 2 ≥ 1 - λ 2 d ∂Ω (x ∞ ) 2 -C 2 η -C 1 /m k = 1 - (λ + ε) 2 d ∂Ω (x ∞ ) 2 + δ. (1.3.16) As x ∞ ∈ M λ+ε (Ω), we have 1 -(λ+ε) 2 d ∂Ω (x∞) 2
≥ 0, and so for any

η ≤ η 0 , k ≥ k 0 and y ∈ B(x ∞ , η), 1 - λ 2 v m k (y) 2 ≥ δ ≥ δ 0 > 0. (1.3.17)
Let (y t ) t≥0 be the curve dened by

y 0 = x ∞ and dy t dt = ∇v m k (y t ). Let T > 0 be such that for any t ∈ [0, T ], d(y t , x ∞ ) ≤ η, and in particular y t / ∈ E m k ,λ . We have v m k (y T ) -v m k (x ∞ ) = (0,T ) |∇v m k (y t )| 2 dt by denition of (y t ), ≥ (0,T ) 1 - λ 2 v m k (y t ) 2 dt because y t / ∈ E m k ,λ ,
≥ T δ because of (1.3.17).

As v m k is bounded by the diameter of Ω, this estimate implies that there exists a nite largest time T > 0 such that for any t ∈

[0, T ], d(y t , x ∞ ) ≤ η. In particular, d(x ∞ , y T ) = η. From Lemma 1.2.
d ∂Ω (y T ) -d ∂Ω (x ∞ ) ≤ |∇d ∂Ω (x ∞ )| d(x ∞ , y T ) + C d (d(x ∞ , y T )) 2 = |∇d ∂Ω (x ∞ )| η + C d η 2 , ≤ 1 - (λ + ε) 2 d ∂Ω (x ∞ ) 2 η + C d η 2 because x ∞ ∈ M λ+ε (Ω). (1.3.18)
In addition,

v m k (y T ) -v m k (x ∞ ) = (0,T ) |∇v m k (y t )| dy t dt dt by denition of (y t ), ≥ (0,T ) 1 - λ 2 v m k (y t ) 2 dy t dt dt because y t / ∈ E m k ,λ , ≥ inf t∈(0,T ) 1 - λ 2 v m k (y t ) 2 η. (1.3.19)

Elastic-plastic torsion problem and λ-medial axis

Using lemma 3.4.1 again, we know that there exists a constant

C 3 > 0 such that d ∂Ω (y T ) - d ∂Ω (x ∞ ) ≥ v m k (y T ) -v m k (x ∞ ) -C 3 m k . Therefore, estimates (1.3.18) and (1.3.19) yield C d η 2 + 1 - (λ + ε) 2 d ∂Ω (x ∞ ) 2 -inf t∈(0,T ) 1 - λ 2 v m k (y t ) 2 η + C 3 m k ≥ 0.
Using (1.3.16), we nd that there exists a constant C 4 > 0 such that for any η ≤ η 0 and k ≥ k 0 ,

1 - (λ + ε) 2 d ∂Ω (x ∞ ) 2 -inf t∈(0,T ) 1 - λ 2 v m k (y t ) 2 ≤ -C 4 .
In particular, for any η ≤ η 0 and k ≥ k 0 , Then, J is called a Jacobi eld if there exists a variation of γ through geodesics whose J is the variation eld, i.e. a smooth map Γ : (-1, 1) × [a, b] → M such that Γ(0, •) = γ and for any s ∈ (-1, 1), Γ(s, •) is a geodesic, and

C d η 2 -C 4 η + C 3 m k ≥ 0.
∂ 1 Γ(0, t) = J(t).
Moreover, if there exists a Jacobi eld J along γ such that J(a) = 0 and J(b) = 0, then we say that γ(a) and γ(b) are conjugate points along γ.

Denition 2.1.2. Let T > 0 and γ : [0, T ] → M be a unit speed geodesic such that γ(0) = b, t 0 ∈ (0, T ) and p = γ(t 0 ). We say that p is a cut point of b along γ if γ is length minimizing between b and p, but not after p, i.e d b (γ(t)) = t for t ≤ t 0 , and d b (γ(t)) < t for t > t 0 . We recall the following. Denition 2.1.4. Let q ∈ M and V ∈ T q M be such that there exists a well dened geodesic γ : [0, 1] → M such that γ(0) = q and γ(0) = V . Then, the (global) exponential map at V is dened by Exp(V ) = γ(1). The restriction of the exponential map Exp on T q M is denoted by exp q and called the exponential map at q.

The following facts can all be found in [74, Chapter III, Section 4]: Proposition 2.1.5. Some well known facts related to the cut locus include:

• Cut b (M ) is the set of points p in M , such that either there exist at least two minimizing geodesics from b to p, or the points b and p are conjugate,

• Cut b (M ) is the closure of the set of points p in M , such that there exist at least two minimizing geodesics from b to p,

• the function d b is smooth outside Cut b (M ) ∪ {b}, and it is dierentiable at a point p if and only if there exist a unique minimizing geodesics between b and p,

• in particular, Cut b (M ) ∪ {b} is the closure of the set of points of non-dierentiability of d b ,

• the exponential map exp b is a dieomorphism from an open set of T b M onto M \ Cut b (M ), • Cut b (M ) is a deformation retract of M \ {b}.
In particular, these two sets have the same homotopy type, and so Cut b (M ) inherits much of the topology of M (like homology groups, for instance). See We know quite a bit about the structure of the cut locus. Let us give some of its properties, to get an idea of what the cut locus looks like. First, it is the union of smooth hypersurfaces consisting of points with exactly two minimizing geodesics, plus a set of Hausdor dimension at most n -2. A reader-friendly reference for this is [START_REF] Itoh | The dimension of a cut locus on a smooth Riemannian manifold[END_REF]. Moreover, the cut locus has locally nite n-1-dimensional Hausdor measure, see for instance [START_REF] Itoh | The Lipschitz Continuity of the Distance Function to the Cut Locus[END_REF]Theorem B]. To our knowledge, the most precise results in this direction are contained in [START_REF] Ardoy | Cut and singular loci up to codimension 3[END_REF], where points of the cut locus are classied, up to a set of Hausdor dimension at most n -3. We refer to [8, Section 1.3] for additional structural properties of the cut locus.

An important property of the distance function, which we will be using in the rest of the thesis, is semiconcavity. Let us give a denition. Denition 2.1.7. Given a constant C > 0, a function u is said to be C-semiconcave on M if and only if for any unit speed geodesic

γ : [a, b] → M , the function t → Ct 2 -u(γ(t)) is convex. We say that u is semiconcave if u is C-semiconcave for some constant C > 0.
We say that u is locally semiconcave if for any p ∈ M , u is semiconcave in a neighborhood of p. Using some regularity result ( [START_REF] Crandall | Viscosity solutions of Hamilton-Jacobi equations in innite dimensions. IV. Hamiltonians with unbounded linear terms[END_REF]) for Hamilton-Jacobi equations, they deduced that for any local chart ψ of M , the map d b • ψ -1 is locally semiconcave as a function on R n (see also [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF] for a more reader-friendly reference about semiconcave functions and Hamilton-Jacobi equations). In the Appendix A.1 about semiconcave functions, we show that it is equivalent to saying that d b is locally semiconcave.

We realized afterwards that a dierent proof of the local semiconcavity of distance functions can also be found in [START_REF] Albano | Singular gradient ow of the distance function and homotopy equivalence[END_REF]. In [START_REF] Mantegazza | On the distributional Hessian of the distance function[END_REF] (and in particular Theorem 2.10), C.Mantegazza, G.Mascellani and G.Uraltsev used this fact along with geometric measure theory tools to give a complete description of the distributional Hessian and Laplacian of d b . They proved the following.

Theorem 2.1.9. The distributional Hessian of d b is given by:

D 2 d b = D 2 d b Vol -(ν ⊗ ν) ∇d + b -∇d - b H n-1 Cut b (M ) ,
where

∆d b = ∆d b Vol -∇d + b -∇d - b H n-1 Cut b (M ) ,
where Ideas of the proof. The proof might seem very technical if the reader is not used to the elementary tools of Riemannian geometry used in the proof, such as normal coordinates, Jacobi elds and covariant derivatives. We will explain the idea of the proof in dimension 2. We will refer to constructions that appear on Figure 2.1. Let p ∈ Cut b (M ). The case where there exist two minimizing geodesics between b and p is simple see below. So let us assume that there exists a unique minimizing geodesic between b and p, along which the two points are conjugate. Let r, R > 0 be such that d b (p) = R + r. For technical reasons, r is to be chosen small. Let p ε a point that is just ε further away than p on the geodesic γ from b to p. Let r ε := r + ε. The barrier we will use is φ ε = R + d( • , ∂B(p ε , r ε )). Let q be the point of γ such that d b (q) = R, so that the circles ∂B(b, R), ∂B(p, r) and ∂B(p ε , r ε ) are all tangent at q. We rst prove that the circles ∂B(b, R) and ∂B(p, r) have the same curvature at q. This is where we use the fact that the points b and p are conjugate. Then, we show that the curvature of ∂B(p ε , r ε ) is larger than the curvature of ∂B(p, r) at q, from which we deduce that the curvature of ∂B(p ε , r ε ) is larger than the curvature of ∂B(b, R) at q. We use this fact to show that we have φ ε ≥ d b in a neighborhood of p. Finally, an easy computation will show that ∆φ ε (p) -→ ε→0 -∞, which will conclude the proof.

We will make heavy use of the notion of exponential map of a hypersurface in the proof.

Let us recall it and x some notations.

Denition 2.2.1. Let S be a smooth oriented hypersurface S of M . Let ν S be the unit normal vector to S. The exponential map of S is dened by

exp S : S × [0, ∞] → M (θ, t) → Exp(tν S (θ))
We will also make heavy use of the notion of normal coordinates. Let us recall this notion and its fundamental properties.

Proposition and denition 2.2.2. For any point p ∈ M , the map exp p : T p M → M is a local dieomorphism from a neighborhood of 0 ∈ T p M to a neighborhood of p ∈ M . Therefore, after identifying T p M with R n by the choice of an orthonormal basis for the metric of M on T p M , the map ψ = exp -1 p is a local chart of M around p, and so it denes some coordinates (x i ). These coordinates are called normal coordinates at p. Proposition 2.2.3. In normal coordinates at p, the tangent vectors ∂ i have vanishing covariant derivatives at p. Moreover, the metric of M is euclidean at p, up to order 1. More precisely, the coordinates of the metric g of M verify

g ij (p) = δ ij and ∂ i g ij (p) = 0.
In particular, recalling the following expression for the Laplacian in coordinates

∆u = 1 √ det g ∂ i ( det gg ij ∂ j ),
one can see that in normal coordinates at p we have ∆u(p) = i ∂ 2 ii u(p). (Please note that the equality holds at p only.)

We will use the following lemma in the proof of the theorem. It is a well known fact when the hypersurface S considered is replaced with a point. We give a proof here, as we could not nd it in the literature. Lemma 2.2.4. Let S be a smooth hypersurface of M such that S is compact. Let µ : [0, r] → M be a geodesic such that q := µ(0) ∈ S. We assume that µ is length minimizing between p := µ(r) and S, and that p is not a cut point of q along µ. Then, the exponential map exp S is a dieomorphism from a neighborhood U (q, r) of (q, r) to a neighborhood U (p) of p, and on U (p) we have

d(•, S) = π 2 • (exp S |U (q,r) ) -1 ,
where π 2 : S × R → R is the projection on the second coordinate. In particular, d(•, S) is smooth around p.

Proof. As p is not a cut point of q along γ, it is not a focal point of S (see [74, Chapter III, lemma 2.11]). In particular, exp S is a local dieomorphism from (q, r) to p. As q is the unique closest point to p on the compact set S, for any neighborhood U S (q) of q in S, there exists a neighborhood U (p) of p such that for any point p ∈ U (p) the closest points to p on S are inside U S (q). We choose such U (p) and U S (q) so that, for some ε > 0, exp S is a dieomorphism from U (q, r) := U S (q) × (r -ε, r + ε) to U (p). Let p ∈ U (p) and r := d(p , S). A closest point to p on S is a point q ∈ U S (q) such that p = exp S (q , r ), and so d(p , S) = r = π 2 • (exp S |U (q,r) ) -1 (p ). This concludes the proof.

Proof of Theorem 2.1.11. Let p ∈ Cut b (M ) and A > 0. We know that either there exist two minimizing geodesics from b to p, or there exists a unique minimizing geodesic from b to p along which the two points are conjugate.

Case one. There exist two minimizing geodesics from b to p. This implies that d b is not dierentiable at p. Let δ > 0 be smaller than the distance between p and Cut p (M ), and ψ : B(p, δ) → R n be a normal coordinate chart around p. According to [60, Proposition 3.4], there exists C > 0 such that the function x → C |x| 2 -d b • ψ -1 (x) is convex. As it is not dierentiable at 0, it has at least two subgradients v and w. We have

C |x| 2 -d b • ψ -1 (x) ≥ max(v • x, w • x) -d b (p), (2.2.1)
with equality for x = 0. For B > 0 to be chosen large enough later, let us dene the function

f : R n → R by f (x) := 1 2 (v • x + w • x) + B(v • x -w • x) 2 -d b (p).
Then, for x in a neighborhood of 0 we have

f (x) = max(v • x, w • x) - 1 2 |v • x -w • x| + B(v • x -w • x) 2 -d b (p) ≤ max(v • x, w • x) -d b (p) (2.2.2)
with equality at x = 0. Setting φ = C |ψ| 2 -f • ψ, we get from (2.2.1) and (2.2.2) that for q in a neighborhood of p, d b (q) ≤ φ(q), with equality at q = p. Moreover, as ψ is a normal coordinate chart, we have

∆φ(p) = ∆(φ • ψ -1 )(0) = 2nC -2B |v -w| 2 .
As B is independent of v, w and A, we may now chose it large enough to have ∆φ(p) ≤ -A, which concludes case one. Case two. There exists a unique minimizing geodesic γ such that γ(0) = b and γ(1) = p.

Let r > 0, and let q be the intersection point of γ with the sphere ∂B(p, r), and R = d b (q). For ε > 0, let us dene

p ε := γ(1 + ε), r ε := r + d(p, p ε ).
We may choose r and ε small enough so that p and p ε are contained in the totally normal geodesic ball B(q, δ) for some δ > 0 (see [START_REF] Carmo | Riemannian geometry[END_REF]Chapter 3,Remark 3.8]). This implies in particular that for any points p and q in neighborhoods of p and q respectively, there exists a unique minimizing geodesic µ (p ,q ) between p and q , and µ (p ,q ) depends continuously on (p , q ). Let us x some normal coordinates at q on B(q, δ) such that the unit normal vector to ∂B(b, R) at q is ∂ n . We will use these normal coordinates throughout the rest of the proof. In these coordinates, around q, ∂B(b, R) is the graph of a smooth function f such that f (0) = 0 and ∇f (0) = 0. Likewise, ∂B(p, r) and ∂B(p ε , r ε ) can be seen around q as the graphs of some smooth functions g and g ε respectively, such that g(0) = g ε (0) = 0 and ∇g(0) = ∇g ε (0) = 0.

Step one. We prove that

∃v ∈ R n-1 , v = 0, v • Hf (0)v = v • Hg(0)v, (2.2.3)
where Hf and Hg are the Hessian matrix of the functions f and g. As the points b and p are conjugate, there exists a Jacobi eld J along γ such that J(0) = 0, J(1) = 0 and J(t) = 0 for t ∈ (0, 1). We may extend it to a global smooth eld J on M . Let (Φ s J ) s∈R be the ow of this vector eld, dened by the following equations:

∀x ∈ M, Φ 0 J (x) = x and ∀s ∈ R, d ds Φ s J (x) = J(Φ s J (x)).
We dene a variation Γ of the curve γ as follows:

∀s ∈ R, ∀t ∈ [0, 1], Γ(s, t) := Φ s J (γ(t)).
For any s ∈ R, Γ(s, •) is still a curve from b to p, as J(0) = 0 and J(1) = 0. Therefore Γ is a xed endpoints variation of γ. As γ is a geodesic, we have

d ds [length(Γ(s, •))] s=0 = 0.
As the variation eld J of Γ is a Jacobi eld, we also have (see [START_REF] Lee | Riemannian manifolds: an introduction to curvature[END_REF]Proposition 10.14] for instance)

d 2 ds 2 [length(Γ(s, •))] s=0 = 0. (2.2.4)
By the implicit function theorem, for any s close enough to 0, there exist some unique times t R (s) and t r (s) such that

d(Γ(s, t R (s)), b) = R and d(Γ(s, t r (s)), p) = r.
Moreover, the functions s → t R (s) and s → t r (s) are smooth. Now we set P (s) := Γ(s, t R (s)) and Q(s) := Γ(s, t r (s)).

We have

length(Γ(s, •)) = length(Γ(s, •) | [0,t R (s)] ) + length(Γ(s, •) | [t R (s),tr (s)] ) + length(Γ(s, •) | [tr (s),1] ) ≥ d(Γ(s, 0), Γ(s, t R (s))) + d(Γ(s, t R (s)), Γ(s, t r (s))) + d(Γ(s, t r (s)), Γ(s, 1)) = R + d(P (s), Q(s)) + r.
Meanwhile, as s goes to 0, thanks to (2.2.4), we also have

length(Γ(s, •)) = length(γ) + o(s 2 ) = R + r + o(s 2 ), so d(P (s), Q(s)) ≤ o(s 2
). In our normal coordinates around q, we have d(x, y) ≥ c |x -y| where c > 0 is a constant and | • | is the euclidean norm. This comes from the fact that the metric g of M and the euclidean metric are locally equivalent in any coordinates system. With this remark, we get

P (s) = Q(s) + o(s 2 ), (2.2.5)
where the equality is to be understood in coordinates. For any s small, let x P (s),

x Q (s) ∈ R n-1
be such that

P (s) = (x P (s), f (x P (s))) and Q(s) = (x Q (s), g(x Q (s))).
The functions x P and x Q are smooth because P and Q are. Using the fact that ∇f (0) = ∇g(0) = 0, (2.2.5) gives

x P (0) • Hf (0)x P (0) = x Q (0) • Hg(0)x Q (0). (2.2.6)
Let us show that x Q (0) = x P (0). We have

P (0) = ∂ s Γ(0, t R (0)) + ∂ t Γ(0, t R (0))t R (0) = J(t R (0)) + γ(t R (0))t R (0).
(2.2.7)

By denition the curve P is on the sphere ∂B(b, R), so P (0) is tangent to this sphere and orthogonal to γ(t R (0)) by the Gauss lemma. Also, as J(0) = 0 and J(1) = 0, the Jacobi eld J is normal to γ (see [56, lemma 10.6]), so J(t R (0)) is orthogonal to γ(t R (0)). Combined with these facts, (2.2.7) yields P (0) = J(t R (0)). Likewise, we have Q (0) = J(t r (0)) = J(t R (0)). So P (0) = Q (0), and consequently x P (0) = x Q (0). Moreover, as J(t) = 0 for t ∈ (0, 1), we have x P (0) = 0. So setting v = x P (0), with (2.2.6) we get (2.2.3).

Step two. Now we want to show that

v • Hg(0)v > v • Hg ε (0)v. (2.2.8)
Let us rst show that g ≥ g ε in a neighborhood of 0. Let us argue by contradiction and assume that for some x ∈ R n-1 , we have g(x) < g ε (x). Let µ : [0, 1] → M be the shortest geodesic from the point of coordinate (x, g(x)) to the point of coordinate (0, r), i.e. p. If x has been taken close enough to 0, then for t ∈ [0, 1], µ(t) stays inside the normal neighborhood of q, B(q, δ), on which our normal coordinates are dened. For t

∈ [0, 1], let x(t) ∈ R n-1 and z(t) ∈ R be such that in coordinates, µ(t) = (x(t), z(t)). We have z(0) = g(x) < g ε (x) = g ε (x(0)). And z(1) = r > 0 = g ε (0) = g ε (x(1)
). As µ is continuous, there exists t ∈ (0, 1) such that

z(t) = g ε (x(t)), i.e. µ(t) ∈ ∂B(p ε , r ε ). This implies r ε = d(µ(t), p ε ) ≤ d(µ(t), p) + d(p, p ε ) < d(µ(0), p) + d(p, p ε ) = r + d(p, p ε ) = r ε ,
which gives a contradiction. We conclude that g ≥ g ε . In particular, for any w ∈ R n-1 , we have w • Hg(0)w ≥ w • Hg ε (0)w. Thus, the matrix Hg(0) -Hg ε (0) is symmetric non-negative.

To show that it is positive denite and conclude that (2.2.8) holds, we only need to show that ∀w ∈ R n-1 , Hg(0)w = Hg ε (0)w.

(2.2.9)

The geodesic γ is minimizing between p ε and ∂B(p ε , r ε ). In particular, the point p is not a cut point of q along γ, so lemma 2.2.4 above shows that the dierential of the map exp ∂B(pε,rε) is invertible at (q, r). On the contrary, the map exp ∂B(p,r) (•, r) is constant and so its dierential is null at q. In particular, for any w ∈ T q ∂B(p, r), we have

D w ν ∂B(pε,rε) (q) = D w ν ∂B(p,r) (q), (2.2.10)
where D w denotes the covariant derivative in the direction w. Let us dene the families of vectors (u g i ) 1≤i≤n-1 and (u gε i ) 1≤i≤n-1 at the point of coordinates

(x, z) ∈ R n-1 × R, by u g i (x, z) := ∂ i + ∂ i g(x)∂ n and u gε i (x, z) := ∂ i + ∂ i g ε (x)∂ n .
These vectors form a basis of the tangent spaces of ∂B(p, r) and ∂B(p ε , r ε ) respectively, as we recall that these surfaces are the graphs of the functions g and g ε . Furthermore, the two basis are identical at (0, 0). Thus,

(2.2.10) is equivalent to ∃i ∈ {1, .., n -1}, u gε i • D w ν ∂B(pε,rε) (q) = u g i • D w ν ∂B(p,r) (q).
Note that if some vector elds X and Y are orthogonal, then we have X • Y = 0, and so

D w X • Y + X • D w Y = 0. So we get ∃i ∈ {1, .., n -1}, D w u gε i • ν ∂B(pε,rε) (q) = D w u g i • ν ∂B(p,r) (q), i.e. ∃i ∈ {1, .., n -1}, D w u gε i • ∂ n = D w u g i • ∂ n .
Recalling that in normal coordinates, the tangent vectors (∂ j ) 1≤j≤n have vanishing covariant derivatives at 0, we get ∃i ∈ {1, .., n -1},

w j ∂ ji g ε (0) = w j ∂ ji g(0),
which is (2.2.9). So (2.2.8) holds.

Step three. Putting steps one and two together, we get

∃v = 0, v • (Hg ε (0) -Hf (0))v < 0.
In particular, there exists a basis

(v i ) 1≤i≤n-1 of R n-1 with v 1 = v, that is orthogonal for the quadratic form q ε (w) = w • (Hg ε (0) -Hf (0))w. Let k ε be a quadratic form of R n-1 such that the (v i ) are orthogonal for k ε , k ε (v) = 0 and for i ≥ 2, k ε (v i ) = max(0, q ε (v i ) + 1)
. This way we have k ε > q ε , and

k ε ≥ 0. Now let us set h ε = g ε -k ε . Note that in dimension
h ε ≤ f on B(0, ρ ε ), for some ρ ε > 0, (2.2.13)
where (2.2.13) comes from the fact that the Hessian of h ε at 0 veries Hh ε (0) = Hg ε (0) -k ε < Hg ε (0) -q ε = Hf (0), and h ε (0) = f (0) = 0, ∇h ε (0) = ∇f (0) = 0. It will be convenient, at the end of the proof, to have a smooth function h : R n-1 → R such that h(0) = 0, ∇h(0) = 0, and for all ε > 0 suciently small, h ε ≥ h.

(

2.2.14)

To see that such function exists, we note that for ε < ε , we have g ε ≥ g ε . This can be proven the same way we proved g ε ≤ g at the beginning of step two. We then x ε > 0. Therefore, for any ε suciently small, we have g ε ≤ g ε ≤ g. This allows us to see that the quadratic forms q ε , and then k ε , are bounded independently of ε, by a quadratic form k. In turn, this implies that the function h = g ε -k veries (2.2.14). Let S hε be the hypersurface of M dened in our normal coordinates around q as the graph of the function h ε on B(0, ρ ε ). Let us set

φ ε = d(•, S hε ) + R.
We will show that for ε suciently small, φ ε is the function φ we are looking for to prove the theorem.

Let us rst show that φ ε ≥ d b in a neighborhood of p. Let p ∈ B(q, δ) be a point near p. Let µ : [0, 1] → M be a length minimizing geodesic between S hε and p . For t ∈ [0, 1], let (x(t), z(t)) ∈ R n-1 × R be the coordinates of µ(t). As µ(0) ∈ S hε , we have z(0) = h ε (x(0)) ≤ f (x(0)). Moreover, the coordinates (x, z) = (0, r) of p verify z > f (x). Therefore, provided p is close enough to p, we have z(1) ≥ f (x(1)). Thus, there exists t ∈ [0, 1] such that

z(t) = f (x(t)), i.e. µ(t) ∈ ∂B(b, R). This implies d(•, S hε ) ≥ d(•, ∂B(b, R)) on a neighborhood of p, (2.2.15)
and so φ ε ≥ d b on a neighborhood of p.

As q ∈ S hε , we also have

φ ε (p) ≤ d(p, q) + R = d b (p). So we get φ ε (p) = d b (p) as well.
We are left to show that φ ε is smooth around p, and that given A > 0, we have ∆φ ε (p) ≤ -A, provided ε has been taken small enough. Let us rst show that φ ε is smooth around p. Between p ε and q, the geodesic γ is minimizing among all geodesics from p ε to ∂B(p ε , r ε ).

The same way we have shown that

h ε ≤ f implies d(•, S hε ) ≥ d(•, ∂B(b, R)) , one can show that h ε ≤ g ε implies d(•, S hε ) ≥ d(•, ∂B(p ε , r ε )) on a neighborhood of p.
(2.2.16) Therefore, between q and p ε , γ is still minimizing among geodesics from p ε to S hε . We may apply lemma 2.2.4 to conclude that φ ε is smooth at p. Now for t in a neighborhood of 0, let us dene c(t) := exp ∂B(pε,rε) (r, exp q (tv, g ε (tv))). We have,

d(c(t), S hε ) ≥ d(c(t), ∂B(p ε , r ε )) because of (2.2.16), = d(c(t), exp q (tv, g ε (tv))) because of lemma 2.2.4, = d(c(t), exp q (tv, h ε (tv))) because of (2.2.12), ≥ d(c(t), S hε ).
Therefore, the rst inequality is actually an equality: d(c(t), S hε ) = d(c(t), ∂B(p ε , r ε )).

As d(•, S hε ) and d(•, ∂B(p ε , r ε )) have the same gradient at c(0) = p, we deduce that

D 2 ċ(0), ċ(0) [d(•, S hε )] p = D 2 ċ(0), ċ(0) [d(•, ∂B(p ε , r ε )] p .
We have ċ(0) = 0 because of lemma 2.2.4 again. Setting

w 1 := ċ(0)/ | ċ(0)|, we get D 2 w 1 ,w 1 [d(•, S hε )] p = D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )] p .
(2.2.17)

Let us complete w 1 into an orthonormal basis (w 1 , .., w n ) of T pM . Let S h be the hypersurface of M dened in our normal coordinates at q as the graph of the function h from (2.2.14).

The same way we have proved (2.2.15), one can prove that d(•, S hε ) ≤ d(•, S h ). As these two functions are equal up to order 1 at p, we deduce that

D 2 w i ,w i [d(•, S hε )] p ≤ D 2 w i ,w i [d(•, S h )] p .
Moreover, we know from [START_REF] Mantegazza | HamiltonJacobi Equations and Distance Functions on Riemannian Manifolds[END_REF]Proposition 3.4] that there exists a constant C > 0 such that for all i, D 2

w i ,w i [d(•, S h )] p ≤ C, so we get ∀i ≥ 2, D 2 w i ,w i [d(•, S hε )] p ≤ C,
where C is independent of ε. Combining this inequality with (2.2.17), we get

∆φ ε (p) ≤ (n -1)C + D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )] p .
(2.2.18)

Let us pick some normal coordinates (x i ) 1≤i≤n at p ε , such that we have ∂ n = γ on the curve γ. By the Gauss lemma, the set of points that are equidistant to ∂B(p ε , r ε ) are orthogonal to the geodesics starting out orthogonally from ∂B(p ε , r ε ). Therefore we have ċ(0) • γ(1) = 0, or equivalently w 1 • ∂ n = 0. In particular, if (w i 1 ) are the coordinates of w relatively to the normal coordinates (x i ) at p ε , we have w n 1 = 0. In these coordinates we have d(x, ∂B(p

ε , r ε )) = r ε -|x|, so D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )](x) = -D 2 w 1 ,w 1 |x| = -w i 1 w j 1 ∂ ij |x| -Γ k ij ∂ k |x| = -w i 1 w j 1 - δ ij |x| + x i x j |x| 3 + Γ k ij (x) x k |x| ,
where the (Γ k ij ) are the Christoel symbols. We apply this formula at the point p of coordinates (0, .., 0, -d(p, p ε )), and obtain

D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )](p) = w i 1 w j 1 δ ij d(p, p ε ) -w i 1 w j 1 Γ n ij (p). (2.2.19)
As we are in normal coordinates, there exist a continuous function ρ : [0, +∞) → [0, +∞) such that ρ(0) = 0 and

Γ k ij (x) ≤ ρ(|x|) and g ij (x) -δ ij ≤ |x| ρ(|x|).
Moreover, note that we only consider coordinates centered at some points p ε contained in a bounded neighborhood of p, so the function ρ is independent of ε. With these remarks, (2.2.19) yields

D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )](p) - |w| 2 d(p, p ε ) ≤ 2 i w i 1 2 ρ(d(p, p ε )).
As ε → 0, the right hand side goes to 0, and

|w| 2 d(p,pε) = 1 d((p,pε)) → +∞, so D 2 w 1 ,w 1 [d(•, ∂B(p ε , r ε )](p) → -∞,
which, together with (2.2.18), concludes the proof.

Chapter 3

Cut locus and variational problem

Introduction

In this chapter, we will use the same notations as in Chapter 2, but our ambient manifold M is assumed to be compact. We recall that n ≥ inf

M |∇u| 2 -mu : u ∈ H 1 (M ), |∇u| ≤ 1, u(b) = 0 , (3.1.1)
where m > 0 is a given constant. We extend the results for the elastic-plastic torsion problem presented in Chapter 1. Our main results are summed up in Theorem 3.1.2 below. For the reader's convenience and in order to make this Chapter relatively self-contained, we will recall some notions and results from the preceding chapters. The elastic-plastic torsion problem is dened as follows:

inf

Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), |∇v| ≤ 1 , (1.1.1)
where Ω is a smooth open subset of R n .

Notice that both the functionals and the constraints in of its minimizer v m have been intensively studied by various authors in the 60's and 70's (see Chapter 1 for some references). In particular, it was proved in [START_REF] Caarelli | The lipschitz character of the stress tensor, when twisting an elastic plastic bar[END_REF]

that v m is locally C 1,1 .
What is more, it was proved in [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF] that the gradient constraint in Indeed, we have

v m ≤ d ∂Ω and v m = arg min Ω |∇v| 2 -mv : v ∈ H 1 0 (Ω), v ≤ d ∂Ω . (3.1.2)
One may ask if an analog of (3.1.2) is also true for u m , with d ∂Ω replaced with d b . Interestingly, it is not. We build a counterexample in Theorem 3.3.1. However, we prove that the equality holds if m is large enough, depending on the Ricci curvature of M . This is Theorem 3.3.2.

The technique of the proof is dierent from the one in [START_REF] Brézis | Équivalence de deux inéquations variationnelles et applications[END_REF], which was specic to the at case.

The medial axis of Ω, denoted by M(Ω), is dened as the set of points of Ω that have at least two closest points on the boundary ∂Ω of Ω:

M(Ω) := x ∈ Ω : ∃y, z ∈ ∂Ω, y = z and d ∂Ω (x) = |x -y| = |x -z| .
Equivalently, M(Ω) is the set of points of Ω at which the distance function d ∂Ω is not differentiable. In [START_REF] Caarelli | The free boundary for elastic-plastic torsion problems[END_REF], the authors proved that for any m > 0, M(Ω) is contained in the set {|∇v m | < 1}, and that the latter converges to the former in the Hausdor sense as m goes to +∞. We prove an analogous result for problem (3.1.1), where the cut locus plays the role of M(Ω), see Theorem 3.1.2 below. One important geometric property of the medial axis M(Ω) is that it is unstable with respect to small perturbations of the boundary of Ω -see Figure 3.1. This instability makes computing numerically M(Ω) quite tricky. Indeed, any numerical approximation of Ω (for instance, with polygons) might introduce an articial (and large) medial set. In order to deal with this problem, in [START_REF] Chazal | The λ-medial axis[END_REF], Chazal and Lieutier dened the so called λ-medial axis of Ω, denoted by M λ (Ω). The latter can be seen to verify

M λ (Ω) = x ∈ Ω : |∇d ∂Ω (x)| 2 ≤ 1 - λ 2 d 2 ∂Ω (x) , (3.1.3)
where ∇d ∂Ω denotes the generalized gradient wherever d ∂Ω is not dierentiable. We have the formula |∇d ∂Ω | (x) = max(0, sup

v∈S n-1 ∂ + v d ∂Ω (x)).
It is proved in [START_REF] Chazal | The λ-medial axis[END_REF] that M λ (Ω) is a good approximation of M(Ω), for λ > 0 small enough. The crucial dierence though is that M λ (Ω) is stable with respect to small variations of the boundary of Ω, whereas M(Ω) is not. We refer the reader to [START_REF] Chazal | The λ-medial axis[END_REF]Section 4] for precise statements and proofs (see also Section 1.2 for more details). This means that although it is dicult to compute numerically M(Ω) directly, one can compute M λ (Ω) instead.

Motivated by these facts, we proved the following in Chapter 1 see Theorem 1.3.1. For λ > 0, let us dene the set E m,λ by

E m,λ := x ∈ Ω : |∇v m (x)| 2 ≤ 1 - λ 2 v 2 m (x)
.

Then, we have

sup x∈E m,λ d(x, M λ (Ω)) -→ m→+∞ 0 and sup x∈M λ+ε (Ω) d(x, E m,λ ) -→ m→+∞ 0, for any ε > 0.
While the medial axis of Ω is unstable with respect to non-smooth variation of the boundary of Ω, the cut locus is unstable with respect to non-smooth variations of the metric of M . Indeed, it is proved in [START_REF] Albano | On the stability of the cut locus[END_REF] that the cut locus is unstable for some C 1 variations of the metric of M (but stable for C 2 variations of the metric of M ). So it is a natural question to try and compute a stable approximation of Cut b (M ). In the following, we prove an analogous result to Theorem 1.3.1 for problem (3.1.1). This result will then be used in Chapter 4 to compute a stable approximation of the cut locus. Before stating it, let us rst dene semiconcavity for functions on a manifold. A function u : M → R is said to be C-semiconcave if for any unit speed geodesic γ, the function t → Ct 2 -u(γ(t)) is convex. We know from [ Following what has been done for the medial axis, we dene a λ-cut locus by

Cut b (M ) λ := x ∈ M : |∇d b (x)| 2 ≤ 1 - λ 2 d 2 b (x)
, for λ > 0.

(3.1.6)

Recall that u m denotes the minimizer of (3.1.1). We set

E m := {x ∈ M : |∇u m (x)| < 1} , and E m,λ := x ∈ M : |∇u m (x)| 2 ≤ 1 - λ 2 u 2 m (x) . (3.1.7)
Our main results are summed up in the following theorem. Theorem 3.1.2. There exists m 0 > 0, depending only on the manifold M , such that for any m > m > m 0 , the minimizer u m of (3.1.1) is locally C 1,1 on M \ {b}, and Given any ε > 0, there exists a constant C > 0 such that, for any m > m 0 ,

Cut b (M ) ⊂ {|∇u m | < 1} ⊂ {|∇u m | < 1}.
u m is C-semiconcave on M \ B(b, ε).

Regularity

Let u d m ('d' is for the 'distance' constraint) be the minimizer of the following strictly convex variational problem: ∆u m ≥ -m.

inf M |∇u| 2 -mu : u ∈ H 1 (M ), u ≤ d b .
This implies, from the theory of subharmonic distributions, that u m has an upper semicontinuous representative. It is proved in [47, section 9], in the present setting of manifolds.

For a reference about subharmonic functions in the euclidean space, see for instance [START_REF] Demailly | Complex Analytic and Dierential Geometry[END_REF].

Let x ∈ Cut b (M ). According to Theorem 2.1.11 we can nd a smooth function φ dened on an open neighborhood N of x such that:

φ ≥ d b , φ(x) = d b (x) and ∆φ ≤ -m -1.
Then the function u m -φ is non-positive strictly subharmonic on N and so by the maximum principle for subharmonic functions, we have

(u m -φ)(x) < 0, i.e. u m (x) < d b (x).
u m ≤ d b -ε ≤ d b,ε = (1 -ρ)d b,ε + ρd b,ε ≤ (1 -ρ)d b + ρd b,ε = d b ,
This concludes the proof.

One could adapt to the manifold framework the regularity theorems for the classical obstacle problem on a euclidean domain and, with the preceding lemma, deduce the regularity of u d m . Rather than doing that, we will use lemma 3.2.1 to reduce our problem to a classical obstacle problem on a euclidean domain. Let us start with the following regularity lemma. inf

Ω |∇u| 2 -mu : u ∈ H 1 (Ω), u ≤ d b , u | ∂Ω = v m| ∂Ω .
As a consequence, u d m -v m is a solution of the following variational problem:

inf

Ω |∇v| 2 -(m + ∆v m )v : v ∈ H 1 0 (Ω), v ≤ d b -v m .
(3.2.5)

Because we have Cut b (M ) ∩ Ω = ∅, the exponential map at b is a dieomorphism onto Ω.

Let φ : Ω → Ω ⊂ R n be a normal coordinates chart centered at b. Let g = (g ij ) denotes the metric of M in the coordinates dened by φ, and det g its determinant. We recall that the Riemannian volume measure is given in coordinates by

√ det gdx. So we have Ω |∇v| 2 -(m + ∆v m )v = Ω g ij ∂ i (v • φ -1 )∂ j (v • φ -1 ) det g -(m + ∆v m ) • φ -1 v • φ -1 det g , so (u d m -v m ) • φ -1 is a minimizer of inf Ω g ij det g∂ i w∂ j w -F w : w ∈ H 1 0 ( Ω), w ≤ ψ , (3.2.6)
where we have set ψ := (d b -v m ) • φ -1 and F := (m + ∆v m ) • φ -1 √ det g. We want to apply [START_REF] Troianiello | Elliptic dierential equations and obstacle problems[END_REF]Theorem 4.32]. For this we need to write the above variational problem into a variational inequality. Let w be a competitor in (3.2.6). Writing down the minimality of

w m := (u d m -v m ) • φ -1 against the competitor w m + t(w -w m ), for t ∈ (0, 1) small, we nd that Aw m , w m -w ≥ F, w m -w ,
where A is the elliptic operator dened on H 1 0 ( Ω) by Aw := -∂ j (g ij √ det g∂ i w). From there, we can apply [START_REF] Troianiello | Elliptic dierential equations and obstacle problems[END_REF]Theorem 4.32] to deduce that, for any p < n, if min(Aψ,

F ) ∈ L p ( Ω), then Aw m ∈ L p ( Ω). To check that min(Aψ, F ) ∈ L p ( Ω), it is enough to check that A(d b • φ -1 ) ∈ L p ( Ω). As d b is smooth except at b, it is enough to check that (A(d b • φ -1 )) p is integrable at 0. But this is a consequence of the fact that -∆d b • φ -1 = 1 √ det g A(d b • φ -1
), and lemma 3.2.5 below, from which we deduce that A(

d b • φ -1 )(x) is equivalent to n-1
|x| when x goes to 0. Therefore, for p < n, (A(d b • φ -1 )) p is integrable at 0, and so Aw m ∈ L p ( Ω). By elliptic regularity, this implies w m ∈ W 2,p ( Ω), for any p < n. By the Sobolev embeddings, w m is then continuous on Ω, and so u d m is continuous on Ω. This concludes the proof.

We

         v ≤ d b in B(b, R), (3.2.7) v = d b in B(b, ε) for some ε ∈ (0, R), (3.2.8) v < 0 in ∂B(b, R), (3.2.9)
∆v ≥ -m in B(b, R) in the distributional sense.

(3.2.10)

We will then show that we have u d m ≥ v. The construction of v is postponed to the end of the proof. From Lemma 3.2.2, we know that the function v -u d m is continuous. Let us rst assume that v -u d m attains a positive maximum at a point x ∈ B(b, R). We have Let us now construct the function v that was used above. Let R > 0 be small enough so that B(b, R) is contained in a normal neighborhood of b. In polar coordinates around b, we dene v as a radial function. For ε > 0 to be chosen small enough later, let f :

0 < v(x) -u d m (x) ≤ d b (x) -u d m (x), so x ∈ E d m . Moreover, we have u d m ≥ 0 since max(u d m , 0) is a better competitor than u d m in (3.2.1), so v -u d m ≤ v < 0 on ∂B(b, R),
[0, R] → [0, ∞) be the C 1 function such that    f (r) = r if ≤ ε, f (r) + n -1 r f (r) = - m 2 if r > ε.
(3.2.12)

If n = 2, the unique C 1 solution to this system is given by:

f (r) = r if r ≤ ε, f (r) = ε + m 8 (ε 2 -r 2 ) + (ε + m 4 ε 2 ) ln( r ε ) if r > ε. (3.2.13) If n = 3, then the solution is          f (r) = r if r ≤ ε, f (r) = ε + m 4n (ε 2 -r 2 ) +(ε n-1 + m 2n ε n ) 1 n -2 ( 1 ε n-2 - 1 r n-2 ) if r > ε. (3.2.14)
Then, we set in standard polar coordinates v(x) = f (r) for x ∈ B(b, R). For r ≤ ε, the constraint (3.2.8) is veried by denition. (For r > ε, we chose f so that ∆v is small, but still larger than -m.)

Let us show that (3.2.7) holds. Let us set g(r) := f (r) -r and prove that g ≤ 0. We have g(r) = 0 for r ≤ ε so it is sucient to prove that g (r) ≤ 0 for r ≥ ε. But, as f veries (3.2.12), g veries

g + n -1 r g = -m - n -1 r for r ≥ ε.
In particular, whenever g (r) = 0, we have g (r) < 0. This implies g (r) ≤ 0 for r ≥ ε, and so (3.2.7) is veried. Now let us show that (3.2.10) holds if R has been taken small enough. We use the following expression of the Laplacian in coordinates:

∆v = 1 √ det g ∂ i ( det gg ij ∂ j v),
where g = (g ij ) is the metric of the manifold M , and det g its determinant. We apply this formula to polar coordinates to nd that, on B(b, R) \ B(b, ε), we have in the classical sense (3.2.17) Furthermore, it is clear from the following expression that f is bounded on [ε, R], by a constant independent of R, as long as we choose R ≤ 1:

∆v = 1 √ det g ∂ r ( det gf (r)) = f + ∂ r det g 2 det g f = f + n -1 r f + ( ∂ r det g 2 det g - n -1 r )f = - m 2 + ( ∂ r det g 2 det g - n -1 r )f .
f (r) = - m n r + (ε n-1 + m n ε n ) 1 r n-1 for ε ≤ r ≤ R.
Hence from (3.2.17) we see that by taking R small enough (independently of ε), we can ensure that ∆v ≥ -m on B(b, R) \ B(b, ε).

But from (3.2.17), we see that the above is also true on B(b, ε) if ε is small enough. Thus the function v is C 1 on B(b, R) and veries ∆v(x) ≥ -m when x / ∈ ∂B(b, ε), hence (3.2.10) holds. It is also clear from (3.2.13) and (3.2.14) that the constraint (3.2.9) is veried if ε is taken small enough. This concludes the proof.

We can now prove a stronger regularity proposition. Proposition 3.2.4. For any ε > 0, the function u d m belongs to C 1,1 (M \ B(b, ε)). Proof. We reproduce the proof of lemma 3.2.2, but we replace the open set Ω with Ω := Ω \ B(b, ε), and the function v m with a function v m that is smooth and such that w m = u m on ∂ Ω. We know that such a function exists because u m is smooth on ∂B(b, ε) for ε small enough, as it can be seen from lemma 3.2.3. This way, we can apply the stronger W 2,∞ regularity result for the obstacle problem [START_REF] Troianiello | Elliptic dierential equations and obstacle problems[END_REF]Theorem 4.38

], since d b is smooth on Ω. We get that u d m belongs to W 2,∞ = C 1,1 ( Ω). As u d m is smooth on E d m and ∂ Ω ⊂ E d m , then u d m is C 1,1 on Ω ∪ E m = M \ B(b, ε).
We end this section with the following elementary computational lemma. Lemma 3.2.5. We have

∆d b (p) = p→b n -1 d b (p) + o(1).
Proof. We compute ∆d b in normal coordinates centered at b. Let g = (g ij ) be the metric of M in these coordinates. We have

∆d b (x) = 1 √ det g ∂ i ( det gg ij ∂ j d b )(x).
In normal coordinates, the metric is euclidean up to order 1 as x goes to 0. So we have

g ij (x) = δ ij + o(x), ∂ i ( det gg ij )(x) = o(1) and 1 √ det g = 1 + o(x).
Furthermore, in normal coordinates, we have d b (x) = |x|, and so

δ ij ∂ ij d b (x) = n -1 |x| .
Putting all the above together, as x goes to 0, we get

∆d b (x) = n -1 |x| + o(1),
which concludes the proof.

Equivalence of the two constraints

The constraints |∇u| ≤ 1 and u(b) = 0, in problem (3.1.1), imply u ≤ d b . However, contrary to the case of the elastic-plastic torsion problem, these two constraints may not be replaced with the constraint u ≤ d b in general. Indeed, we have the following proposition. We remind the reader that u m and u d m denotes the minimizer of problems (3.1.1) and (3.2.1) respectively.

Theorem 3.3.1. There exists a surface of revolution M and a parameter m > 0 such that

u m = u d m .
However, when m is large enough, we do have u d m = u m : Theorem 3.3.2. Let K ≥ 0 be such that

Ric ≥ -K, (3.3.1)
where Ric denotes the Ricci curvature tensor of the manifold M . If

m ≥ 1 2 max( nK(1 + Kdiam(M ) 2 ), nKdiam(M )),
then ∇u d m ≤ 1, or equivalently

u d m = u m .
Let us now prove these two theorems.

Proof of Theorem 3.3.1. Let r θ denote the rotation of R 3 of angle θ ∈ [0, 2π) around the z-axis.

Let T := 10 10 and r, h : [0, T ] → R be two smooth functions such that γ : t → (r(t), 0, h(t)) is a unit speed curve.

M := {r θ (γ(t)) : (t, θ) ∈ [0, T ] × [0, 2π)]} is a smooth surface, r(0) = 0, r([1, 2]) ⊂ [1, +∞), r([3, 4]) ⊂ (0, 10 -10 ), r([5, T -1]) ⊂ [1, +∞).
This information is pictured in Figure 3.2. We chose b = (0, 0, 0) as the base point on M , and m = 10 -10 . Let us assume that u d m = u m and build a better competitor in (3.2.1) to contradict the minimality of u d m . We will rst reduce (3.2.1) to a one-dimensional problem. Note that the functional we are minimizing is rotation-invariant. More precisely, for any θ ∈ (0, 2π) and u ∈ H 1 (M ), we have

M |∇(u • r θ )| 2 -m(u • r θ ) = M |∇u| 2 -mu. (3.3.2)
By uniqueness of the minimizer u d m , we deduce that u d m is rotation-invariant, i.e. there exists a function ρ m : [0, T ] → R such that for any θ ∈ [0, 2π) and t ∈ [0, T ], u d m (r θ (γ(t))) = ρ m (t). Thus u d m is a minimizer of (3.2.1) among rotation-invariant functions. Let u : M → R be any rotation-invariant function, and ρ : [0, T ] → R be such that for any θ ∈ [0, 2π),

x z γ(0) • γ(1) • γ(1) • γ(2) • γ(3)• γ(4)• γ(5) • γ(T -1) • γ(T )• r(t)
•γ(t) u(r θ (γ(t))) = ρ(t). We will translate the minimization problem (3.2.1) on u into a problem on ρ.

First, because M is a surface of revolution, all the geodesics starting from b = (0, 0, 0) have a constant angle θ (this is an exercise in Riemannian geometry). Thus, they are of the form t → r θ (γ(t)) for some θ ∈ [0, 2π). These are actually unit speed geodesics as γ is unit speed. Hence, d b (r θ (γ(t))) = t, and the constraint u ≤ d b in (3.2.1) is equivalent to ρ(t) ≤ t.

Secondly, we translate the H 1 constraint. To this end, let us dene some coordinates (t, θ) on M via the map φ : (0, T ) × (0, 2π) → M (t, θ) → r θ (γ(t)) .

We have

M |∇u| 2 = (0,T )×(0,2π) (|∇u| 2 • φ)Jφdtdθ = (0,T )×(0,2π) |∇u| 2 (r θ (γ(t)))r(t)dtdθ = 2π (0,T ) |∇u| 2 (γ(t))r(t)dt, (3.3.3)
because u is rotation-invariant. Furthermore, as u is rotation-invariant, its gradient at the point γ(t) is parallel to γ (t), and so

ρ (t) = ∇u(γ(t)) • γ (t) = |∇u(γ(t))| γ (t) = |∇u(γ(t))| . Hence (3.3.3) gives M |∇u| 2 = 2π (0,T ) ρ (t) 2 r(t)dt.
Thus, the constraint u ∈ H 1 (M ) in (3.2.1) is equivalent to v ∈ H 1 ((0, T ), r(t)dt).

Thirdly, we may compute the functional likewise:

M |∇u| 2 -mu = 2π (0,T ) ρ (t) 2 -mρ(t) r(t)dt.
Putting these facts together, as u d m is a minimizer in (3.2.1), we deduce that ρ m is a minimizer of :

inf (0,T ) ρ (t) 2 -mρ(t) r(t)dt : ρ ∈ H 1 ((0, T ), r(t)dt), ρ(t) ≤ t . (3.3.4)
The idea of the rest of the proof is the following. First, we recall the assumption u d m = u m , which means that ∇u d m ≤ 1, and so |ρ m | ≤ 1. Now, if ρ m (4) is close to 4, then ρ m (t) is close to 1 for t ≤ 4, so a competitor v such that ρ (t) is small for t ≤ 4 will contradict the minimality of ρ m in (3.3.4). If on the contrary ρ m (4) is signicantly smaller than 4, then for t ≥ 4, ρ m (t) will be signicantly smaller than t, so a competitor ρ such that ρ(t) is closer to t for t ≥ 4 will contradict the minimality of ρ m in (3.3.4). Because we chose r very small on the interval [START_REF] Alama | Droplet breakup in the liquid drop model with background potential[END_REF][START_REF] Albano | On the stability of the cut locus[END_REF] (see Figure 3.2), we can dene a competitor ρ independently on [0, 3] and [4, T ], without paying much for the behavior of ρ on [START_REF] Alama | Droplet breakup in the liquid drop model with background potential[END_REF][START_REF] Albano | On the stability of the cut locus[END_REF].

Case one: ρ m (4) ∈ [3.5, 4]. Let us dene a competitor ρ for (3.3.4):

ρ : [0, T ] → R t →      0 if t ∈ [0, 3], 4(t -3) if t ∈ [3, 4] ρ m (t) + 4 -ρ m (4) if t ≥ 4,
Let us call F(ρ) the functional appearing in (3.3.4). We have, from the denition of r and ρ,

F(ρ) = (3,4) (16 -4m(t -3)) r(t)dt + (4,T ) ρ 2 m (t) -mρ m (t) r(t)dt -m(4 -ρ m (4)) (4,T ) r(t)dt ≤ (16 -0) • 10 -10 + (4,T ) ρ 2 m (t) -mρ m (t) r(t)dt -0 (3.3.5) so F(ρ) -F(ρ m ) ≤ 16 • 10 -10 - (0,4) ρ 2 m (t) -mρ m (t) r(t)dt ≤ 16 • 10 -10 - (1,2) ρ 2 m (t)r(t)dt + m (0,4) ρ m (t)dt ≤ 16 • 10 -10 - (1,2) ρ 2 m (t)r(t)dt + m (0,4) tdt = 16 • 10 -10 - (1,2) ρ 2 m (t)
ρ m ≤ (1,2) 1 r 1 2
(1,2)

ρ 2 m r 1 2
, and so Case two: ρ m (4) ≤ 3.5. We use the same competitor ρ as in case one. We even perform similar estimates, the only dierence being that we do not estimate the term -m(4ρ m (4)) (4,T ) r(t)dt by 0 as in (3. To estimate the Laplacian of d 2 p , we use some normal coordinates (x i ) centered at p. In these coordinates, the metric is euclidean up to order 1, uniformly in p since M is compact, and d p (x) = |x|. This argument has been detailed in the proof of lemma 3.2.5, so we do not reproduce it here. We get that for m large enough, independently of p, Proof. The fact that, for any m > 0, Cut b (M ) ⊂ E d

(1,2) ρ 2 m r ≥ (ρ m (2) -ρ m (1)) 2 (1,2) 1 r ≥ (ρ m (2) -ρ m (1)) 2 , (3.
∀p ∈ B 5n/m (p), ∆d 2 p (p) ≤ 2(2n).

Semiconcavity

In this section, we prove (3.1.10) and (3.1.11) of Theorem 3.1.2. We begin with a denition.

For a, b, λ ∈ R, we set λ ab = (1 -λ)a -λb.

Denition 3.5.1. Given a constant C > 0, a function u is said to be C-semiconcave on M if and only if for any unit speed geodesic γ : [a, b] → M , the function t → Ct 2 -u(γ(t)) is convex. This is equivalent to having, for any λ ∈ [0, 1],

(

1 -λ)u(γ(a)) + λu(γ(b)) -u(γ(λ ab )) ≤ Cλ(1 -λ)(b -a) 2 .
We say that u is semiconcave if u is C-semiconcave for some constant C > 0. We say that u is locally semiconcave if for any p ∈ M , u is semiconcave in a neighborhood of p.

Remark 3.5.2. On the compact manifold M , semiconcavity and local semiconcavity are the same.

In [60, Denition 2.5], the authors dene local semiconcavity using charts as follows: a function u is said to be locally semiconcave if for any chart ψ, the function u • ψ -1 is locally semiconcave as a function on R n . In the appendix, Proposition A.1.4, we show that this notion of local semiconcavity is equivalent to the denition of local semiconcavity we gave.

In [60, Proposition 3.4], the authors proved that the distance function is locally semiconcave on M \ {b}. As a consequence, recalling remark 3.5.2, we have the following proposition: Proposition 3.5.3. Given ε > 0, the distance function d b is C-semiconcave on M \ B(b, ε), for some C > 0.

We also have the following obvious proposition, obtained from the caracterization of twice dierentiable convex functions on the real line: Proposition 3.5.4. A C 2 function u : M → R is C-semiconcave if and only if its second order covariant dierential veries D 2 u ≤ 2C.

We can now prove the following: Proposition 3.5.5. Given ε > 0, there exists a constant C > 0 such that for any m large enough, the function u m is C-semiconcave on M \ B(b, ε).

We start with a proof that is specic to the dimension 2. It uses the determinant of symmetric matrices of size 2, to estimate their eigenvalues. We will give a proof that works in any dimension afterwards. The rst part of the proof is almost identical to the rst part of the proof of Theorem 1.3.2.

Proof of Proposition 3.5.5 in dimension 2. The reader may nd the recalls of appendix A.1 useful to follow this proof. For m large enough, we recall that u m = u d m (Theorem 3.3.2). This will be used throughout the proof. Let C d be such that the distance function d b is C dsemiconcave on M \ B(b, ε). We will show that u m is (C d + 1)-semiconcave for m large enough. Let us introduce some notations. For a, b ∈ R and λ ∈ (0, 1),

λ ab := (1 -λ)a + λb.
g ab D a D b D c D d u m ≥ -C M |∇u m | -C M D 2 u m -g ab D c D d D a D b u m .
In E m , u m has constant Laplacian, so

g ab D c D d D a D b u m = D c D d g ab D a D b u m = D c D d ∆u m = 0. So we get g ab D a D b D c D d u m ≥ -C M |∇u m | -C M D 2 u m .
∆( D 2 u m 2 ) ≥ -C M |∇u m | -C M D 2 u m D 2 u m ≥ -C M |∇u m | 2 -C M D 2 u m 2 .
Coming back to (3.5.10), we get:

∆(det D 2 u m ) ≤ C M |∇u m | 2 + C M D 2 u m 2 .
(3.5.17)

To get a function with non-positive Laplacian, we need to add some terms to det D 2 u m to get a non-positive right hand side in (3.5.17). First we use the Bochner-Weitzenböck formula

(3.3.10) to get ∆(|∇u m | 2 ) ≥ D 2 u m 2 -C M |∇u m | 2 .
(3.5.18) With (3.5.17), we obtain

∆(det D 2 u m -C M |∇u m | 2 ) ≤ C M |∇u m | 2 . (3.5.19)
Secondly, we use the equality

∆(u 2 m ) = 2 |∇u m | 2 + 2u m ∆u m = 2 |∇u m | 2 -2mu m , to get ∆(u 2 m ) ≥ 2 |∇u m | 2 -C M m,
∆(det D 2 u m -C M |∇u m | 2 -C M u 2 m ) ≤ C M m,
Finally, we use the equation ∆u m = -2m to get that for m large enough,

∆(det D 2 u m -C M |∇u m | 2 -C M u 2 m + C M u m ) ≤ 0.
(3.5.21)

Now that we have a superharmonic function on E m , we need to look at its boundary values to apply the maximum principle. Thus we need to control the second order derivatives of u m near the boundary of E m . As u m is not C 2 across ∂E m , we cannot directly use the fact that the hessian of d b is bounded on ∂E m . We will use a theorem for obstacle problems on R n . Let us show that u m is the solution of an obstacle problem on an open set of R 2 . Then, we will apply [START_REF] Friedman | Variational principles and free-boundary problems[END_REF]Theorem 3.8] Let Ω be dened as in the proof of 3.2.4. Let φ : Ω → U be a normal coordinates chart.

Writing down (3.5.23) in these coordinates, we nd

A u m -2m ≥ 0, u m ≤ ψ and (A u m -2m)( u m -ψ) = 0,
where A is the Laplacian of M in the coordinates dened by φ,

u m = u m •φ -1 and ψ = d b •φ -1 .
This is the form of [37, Chapter 2, equation (3.16)], so we can apply [37, Chapter 2, Theorem

3.8], to deduce that ∀p ∈ ∂E m , ∀X ∈ R 2 lim q→p q∈Em D 2 u m (φ(q))(X, X) ≤ D 2 ψ(φ(p))(X, X).
(3.5.24)

Moreover, we have

D 2 u m = D 2 u m • (Dφ -1 , Dφ -1 ) + Du m • D 2 φ -1 , D 2 ψ = D 2 d b • (Dφ -1 , Dφ -1 ) + Dd b • D 2 φ -1 ,
and Du m = Dd b on ∂E m because u m is C 1 . Thus, (3.5.24) yields:

∀p ∈ ∂E m , ∀X ∈ R 2 lim q→p q∈Em D 2 u m (q)(X q , X q ) ≤ D 2 d b (p)(X p , X p ),
where we have set X q := Dφ -1 (φ(q))X.

As d b is C d -semiconcave, with Proposition 3.5.4, we get ∀p ∈ ∂E m , ∀X ∈ R 2 lim q→p q∈Em D 2 u m (q)(X q , X q ) ≤ C d |X p | 2 .
(3.5.25)

From there, we deduce that for q ∈ E m close enough to ∂E m , we have

D 2 u m (q) ≤ C d + 1/4.
(3.5.26) Indeed, if not, there exist a sequence (q k ) of points of E m whose distance to ∂E m goes to 0, and a sequence (X k ) of unit vectors of R 2 such that for any k ∈ N,

D 2 u m (q k )((X k ) q k , (X k ) q k ) > C d + 1 4 . (3.5.27)
As E m is precompact, up to extracting a subsequence, we can assume that (q k ) converges to a point p ∈ ∂E m , and (X k ) converges to a vector Y ∈ R 2 . Because of (3.5.25), we have

lim k→∞ D 2 u m (q k )(Y q k , Y q k ) ≤ C d .
(3.5.28) Furthermore, we know from proposition 3.2.4 that D 2 u m is locally bounded. As (X k ) q k -Y q k converges to 0 when k goes to ∞, this implies Let q be such that (3.5.26) holds. Let λ ≤ µ be the eigenvalues of D 2 u m (q). The inequality

lim k→∞ D 2 u m (q k )((X k ) q k , (X k ) q k ) -D 2 u m (q k )(Y q k , Y q k ) = 0.
det D 2 u m (q) = λµ = -µ 2 -2mµ ≥ -2(C d + 1 2 )m.
Recalling that |∇u m | ≤ 1 (Theorem 3.3.2) and that u m is bounded by the diameter of M , we deduce that for m large enough (depending on M only), at any point q ∈ E m suciently close to ∂E m ,

det D 2 u m -C M |∇u m | 2 -C M u 2 m + C M u m ≥ -2(C d + 1 2 )m -C M ≥ -2(C d + 3 4 )m.
Recalling (3.5.21), by the maximum principle on E m , we get

det D 2 u m -C M |∇u m | 2 -C M u 2 m + C M u m ≥ -(C d + 3 4 )m.
Using the boundedness of the terms ∇u m and u m once more, we nd that inside E m and for m large enough, (3.5.9) is valid. This concludes the proof. Now let us give a proof of Proposition 3.5.5 that works in any dimension. This technique is inspired from the proof of second order estimate for hessian equations in Riemannian manifolds by Guan in [START_REF] Guan | Second Order Estimates and Regularity for Fully Nonlinear Elliptic Equations on Riemannian Manifolds[END_REF]. The reader may nd the recalls of appendix A.2 useful to follow this proof.

We will use many arguments and estimates from the proof of the 2-dimensional case.

Proof of Proposition 3.5.5. As in the preceding proof, we show that it is enough to prove that for m large enough, u m is C-semiconcave in E m , where C is a constant that depends only on M . Let C 1 , C 2 , C 3 > 0 be some constants to be taken large enough later. For p ∈ E m and X ∈ S n-1 (T p M ), let us dene

f (p, X) := D 2 u m (X, X) + C 1 |∇u m | 2 (p) + C 2 u 2 m (p) -C 3 u m (p).
(3.5.32)

We will show that for a good choice of constant C i depending only on M , f is bounded by a constant that depends only on M . This will conclude the proof since |∇u m | 2 , u 2 m and u m are also bounded by a constant depending only on M . Let us assume that there exist q ∈ E m and Y ∈ S n-1 (T q M ) such that f (q, Y ) = sup p∈Em X∈S n-1 (TpM ) f (p, X), (3.5.33) and show that it gives a contradiction. Let us pick some normal coordinates at q such that ∂ 1 (q) = Y . We then extend the vector Y into a vector eld (still denoted by Y ) in a neighborhood of q, by setting Y := ∂ 1 / |∂ 1 |. As D∂ 1 (q) = 0, we also have DY (q) = 0. Moreover, as the manifold is compact, D 2 Y (q) is bounded by a constant that depends on M only. In the following, any constant that depends on M only will be denoted by C M . We have

D 2 Y (q) ≤ C M . (3.5.34)
We will show that the Laplacian of p → f (p, Y (p)) is positive at q, which contradicts the maximality of (q, Y (q)) in (3.5.33). Let us estimate ∆(D 2 u m (Y, Y )) at the point q.

∆(D 2 u m (Y, Y )) = g ab D a D b (D 2 cd u m Y c Y d ) = g ab D 4 abcd u m Y c Y d + D 3 acd u m D b (Y c Y d ) +D 3 bcd u m D a (Y c Y d ) + D 2 cd u m D 2 ab (Y c Y d ) .
g ab D a D b D c D d u m ≥ -C M |∇u m | -C M D 2 u m .
From this and the fact Y has norm 1, we deduce

g ab D a D b D c D d u m Y c Y d ≥ -C M -|∇u m | 2 -D 2 u m 2 .
We obtain, at the point q,

∆(D 2 u m (Y, Y )) ≥ -C M -D 2 u m 2 -|∇u m | 2 .
(3.5.35)

We recall the following from the preceding proof:

∆ |∇u m | 2 ≥ D 2 u m 2 -C M |∇u m | 2 , (estimate (3.5.18)) ∆u 2 m ≥ |∇u m | 2 -C M m, (estimate (3.5.20)) ∆u m = -2m.
Using these three estimates and (3.5.35), we nd some constants

C M 1 , C M 2 , C M 3 > 0 such that ∆(D 2 u m (Y, Y ) + C M 1 |∇u m | 2 + C M 2 u 2 m -C M 3 u m ) > 0.
So, in the denition of f (3.5.32), if we take

C i = C M i for i = 1, 2, 3, we get ∆(f (p, Y (p))) p=q > 0,
which contradicts the maximality of (q, Y (q)). Therefore the supremum in (3.5.33) is not attained in E m . We conclude as in the preceding proof by showing that, by the semiconcavity of d b , f (p, X) is bounded by a constant independent of m when p gets close to ∂E m and X ∈ S n-1 (T p M ).

Now, putting all sections together, only (3.1.11) remains to be proven in Theorem 3.1.2. This is the object of the following proposition.

lim inf k→∞ |∇u m k (p k )| ≥ |∇d b (p ∞ )| .
Using the facts that p k ∈ E m k ,λ and u m k converges uniformly to d b , this implies

|∇d b (p ∞ )| 2 ≤ 1 - λ 2 d b (p ∞ ) 2 ,
and so p ∞ ∈ Cut b (M ) λ . But this contradicts (3.5.38). Thus (3.5.36) is proved.

For the proof of (3.5.37), once again we argue by contradiction and assume that it does not hold. Let (m k ) k≥0 and (p k ) k≥0 be sequences of (0, ∞) and M respectively, such that m k -→ k→+∞ +∞ and for some η > 0, for any k ≥ 0, p k ∈ Cut b (M ) λ+ε and d(p k , E m k ,λ ) > 2η. (3.5.39) Up to taking subsequences, we can assume that (p k ) converges to a point p ∞ in Cut b (M ) λ+ε . The last inequality implies that for any k large enough, we have

d(p ∞ , E m k ,λ ) > η.
According to lemma 3.4.1, there exists a constant C 1 > 0 such that for any p ∈ M and k ≥ 0,

u m k (p) 2 ≥ d b (p) 2 -C 1 /m k . As d b is 1-Lipschitz and bounded, we deduce that there exists a constant C 2 > 0 such that for any p ∈ B(p ∞ , η) and k ≥ 0, u m k (p) 2 ≥ d b (p ∞ ) 2 -C 2 η -C 1 /m k .
Let η 0 > 0, k 0 > 0 and δ 0 > 0 be constants such that for any η ≤ η 0 and k ≥ k 0 ,

δ := (λ + ε) 2 d b (p ∞ ) 2 - λ 2 d b (p ∞ ) 2 -C 2 η -C 1 /m k ≥ δ 0 > 0.
For any p ∈ B(p ∞ , η), we have

1 - λ 2 u m k (p) 2 ≥ 1 - λ 2 d b (p ∞ ) 2 -C 2 η -C 1 /m k = 1 - (λ + ε) 2 d b (p ∞ ) 2 + δ. (3.5.40) As p ∞ ∈ Cut b (M ) λ+ε , we have 1 -(λ+ε) 2 d b (p∞) 2
≥ 0, and so for any η ≤ η 0 , k ≥ k 0 and p ∈ B(p ∞ , η),

1 - λ 2 u m k (p) 2 ≥ δ ≥ δ 0 > 0. (3.5.41)
Let (q t ) t≥0 be the curve dened by q 0 = p ∞ and dq t dt = ∇u m k (q t ).

Let T > 0 be such that for any t ∈ [0, T ], d(q t , p ∞ ) ≤ η, and in particular q t / ∈ E m k ,λ . We have

u m k (q T ) -u m k (p ∞ ) = (0,T )
|∇u m k (q t )| 2 dt by denition of (q t ),

≥ (0,T ) 1 - λ 2 u m k (q t ) 2 dt because q t / ∈ E m k ,λ , ≥ T δ because of (3.5.41).
As u m k is bounded by the diameter of M , this estimate implies that there exists a nite largest time T > 0 such that for any t ∈ [0, T ], d(q t , p ∞ ) ≤ η. In particular, d(p ∞ , q T ) = η. Let γ be a unit speed minimizing geodesic between p ∞ and q T . As d b is semiconcave (Proposition 3.5.3), there exists a constant C d > 0 such that the function f :

t → C d t 2 -d b (γ(t)) is convex.
In particular, we have f (0

) -f (d(p ∞ , q T )) ≤ -f (0)d(p ∞ , q T ), which yields d b (q T ) -d b (p ∞ ) ≤ |∇d b (p ∞ )| d(p ∞ , q T ) + C d (d(p ∞ , q T )) 2 = |∇d b (p ∞ )| η + C d η 2 , ≤ 1 - (λ + ε) 2 d b (p ∞ ) 2 η + C d η 2 because p ∞ ∈ Cut b (M ) λ+ε . (3.5.42)
In addition,

u m k (q T ) -u m k (p ∞ ) = (0,T )
|∇u m k (q t )| dq t dt dt by denition of (q t ),

≥ (0,T ) 1 - λ 2 u m k (q t ) 2 dq t dt dt because q t / ∈ E m k ,λ , ≥ inf t∈(0,T ) 1 - λ 2 u m k (q t ) 2 η. (3.5.43)
Using lemma 3.4.1 again, we know that there exists a constant

C 3 > 0 such that d b (q T ) - d b (p ∞ ) ≥ u m k (q T ) -u m k (p ∞ ) -C 3 m k
. Therefore, estimates (3.5.42) and (3.5.43) yield

C d η 2 + 1 - (λ + ε) 2 d b (p ∞ ) 2 -inf t∈(0,T ) 1 - λ 2 u m k (q t ) 2 η + C 3 m k ≥ 0.
Using (3.5.40), we nd that there exists a constant C 4 > 0 such that for any η ≤ η 0 and k ≥ k 0 ,

1 - (λ + ε) 2 d b (p ∞ ) 2 -inf t∈(0,T ) 1 - λ 2 u m k (q t ) 2 ≤ -C 4 .
In particular, for any η ≤ η 0 and k ≥ k 0 ,

C d η 2 -C 4 η + C 3 m k ≥ 0.

λ-Cut locus

In [START_REF] Dey | Cut locus and topology from surface point data[END_REF], the authors computed shortest curves on a graph obtained from a suciently dense sample of points of a surface. From there they deduced an approximation of the cut locus, and ltered it according to the maximal distance (called spread) between the geodesics meeting at a point of the approximated cut locus. They proved that the set they compute converges to the cut locus (see [START_REF] Dey | Cut locus and topology from surface point data[END_REF]Theorem 4.1]).

We also mention [START_REF] Bonnard | Geometric and numerical techniques to compute conjugate and cut loci on riemannian surfaces[END_REF], where the authors use some more geometric tools to compute (numerically) the cut locus of an ellipsoid, or a sphere with some particular metric with singularities.

Our method. The strategy we use is quite dierent. Given m > 0 a constant, let u m be the minimizer of the following variational problem inf

S |∇ S u| 2 -mu : u ∈ H 1 (M ), |∇ S u| ≤ 1, u(b) = 0 , (4.1.1)
where ∇ S denotes the gradient operator on the surface S. For λ > 0 to be chosen small, we will

use the set E m,λ := |∇ S u m | 2 ≤ 1 -λ 2 u 2 m
as an approximation of Cut b (S). This is justied by Section 4.2, plus the theoretical results regarding problem (4.1.1) obtained in Chapter 3, which will be summarized in Section 4.3. Now the set E m,λ can be well approximated using nite elements on a triangulation of the surface S.

The rest of the Chapter is organized as follows. In Section 4.2, we recall the denition of "λ-cut locus" Cut b (S) λ and show that it can be used as an approximation of the compute cut locus. In Section 4.3, we recall the results from Chapter 3 which states that the set E m,λ dened above is a good approximation of Cut b (S) λ if m is large enough. In Section 4.4, we discretize problem (4.1.1) using nite elements and prove that the "discrete E m,λ " converges to E m,λ . In Section 4.5, we present the results of some numerical experiments.

λ-Cut locus

We recall that, according to [START_REF] Mantegazza | HamiltonJacobi Equations and Distance Functions on Riemannian Manifolds[END_REF]Proposition 3.4], the function d b is locally semiconcave on S \ {b}, so it has a generalized gradient everywhere on S \ {b}, whose norm is given by the following formula (see propositions A.1.9 and A.1.6):

|∇ S d b | (x) = max(0, sup v∈TxS,|v|=1 ∂ + v d b (x)).
We recall the identity (3.1.5):

Cut b (M ) = {|∇d b | < 1}.
In Section 3.1, following the denition of the λ-medial axis by Chazal and Lieutier, we dene the λ-cut locus as

Cut b (S) λ := x ∈ S \ {b} : |∇ S d b (x)| 2 ≤ 1 - λ 2 d 2 b (x)
,

where ∇d b denotes the generalized gradient of d b wherever it is not dierentiable. The following obvious proposition holds. In addition, we have the following proposition.

Proposition 4.2.2. If S is a real analytic surface, then for λ > 0 small enough, one of the connected component of Cut b (S) λ has the same homotopy type as Cut b (S), while the other connected components, if any, are contractible.

These two propositions justify that Cut b (S) λ is a good approximation of Cut b (S), for λ > 0 small enough.

Using Proposition A.1.10, Proposition 4.2.2 will mainly be a consequence of [START_REF] Dey | Cut locus and topology from surface point data[END_REF]Proposition 3.4] and the proof of [START_REF] Dey | Cut locus and topology from surface point data[END_REF]Proposition 3.5]. Following [START_REF] Dey | Cut locus and topology from surface point data[END_REF], we will use the following terminology. A point x of a nite graph G is called a tree point if G \ {x} has a connected component whose closure is a tree. Otherwise, x is called a cycle point. It is a consequence of the proof of [START_REF] Dey | Cut locus and topology from surface point data[END_REF]Proposition 3.5] that any closed connected subset of G that contains all cycle points is a deformation retract of G.

Proof of Proposition 4.2.2. As S is real analytic, the cut locus Cut b (S) is a nite graph (see [START_REF] Myers | Connections between dierential geometry and topology ii. closed surfaces[END_REF] in dimension 2, and [START_REF] Buchner | Simplicial structure of the real analytic cut locus[END_REF] for the generalization to arbitrary dimensions). According to the Proposition A.1.10, given any θ > 0, if λ has been taken small enough, then for any point x ∈ Cut b (S) \ Cut b (S) λ , the angle between the minimizing geodesics from b to x, at the point x, is smaller than θ. Given two unit speed minimizing geodesics γ 1 and γ 2 , following [START_REF] Dey | Cut locus and topology from surface point data[END_REF], the spread between γ 1 and γ 2 is dened as Therefore, we will use Cut b (S) λ as an approximation of Cut b (S) for λ small enough.

spd(γ 1 , γ 2 ) = sup t d(γ 1 (t), γ 2 (t)).

Approximation with a variational problem

For m > 0, recall that u m is the minimizer in (4.1.1). For λ > 0, let us dene the set E m,λ by

E m,λ := x ∈ S \ {b} : |∇ S u m (x)| 2 ≤ 1 - λ 2 u 2 m (x)
.

We recall the following theorem from Chapter 3.

Discretization

Finite elements of order r on a surface approximation of order k

In this section we introduce a discretization framework adapted to the variational problem (4.1.1) based on nite elements. We follow the notations of [START_REF] Demlow | Higher-order nite element methods and pointwise error estimates for elliptic problems on surfaces[END_REF][START_REF] Dziuk | Finite element methods for surface pdes[END_REF].

Let S be a compact oriented smooth two-dimensional surface embedded in R 3 . For x ∈ S, we denote by ν(x) the oriented normal vector eld on S. Let d : R 3 → R be the signed distance associated to S and U η = {x ∈ R 3 , |d(x)| < η} the tubular neighborhood of S of width η > 0.

It is well known that if η is small enough (for instance 0 < η < min i=1,2

1 |κ i | L∞(S)
where the (κ i ) stand for the extremal sectional curvatures of S), then for every x ∈ U η it exists a unique a(x) ∈ S such that x = a(x) + d(x)ν(a(x)) = a(x) + d(x)∇d(x). We consider S 1 h a triangular approximation of S whose vertices lie on S and whose faces are quasi-uniform and shape regular of diameter at most h > 0. Moreover, we will assume that T h , the set of triangular faces of S h , are contained in some tubular neighborhood U η such that the map a dened by (4.4.1) is unique.

For k ≥ 2 and for a triangle T ∈ T h , we consider the n k Lagrange basis functions Φ k 1 , . . . Φ k n k of degree k and dene the discrete projection on S h by:

a k (x) = n k j=1 a(x j )Φ k j (x) (4.4.2)
where x 1 , . . . , x n k are the nodal points associated to the basis functions. Now we can dene S k h a polynomial approximation of order k of S associated to T h

S k h = {a k (x), x ∈ S h }. (4.4.3)
Observe that by denition the image by a of the nodal points are both on S and on S k h . Let us now introduce the nite element spaces on S h = S 1 h and S k h for k ≥ 2. For every integer r ≥ 1, let

L r h = {χ ∈ C 0 (S h ), χ| T ∈ P r , ∀T ∈ T h } (4.4.4)
where P r is the family of polynomials of degree at most r.

Analogously, for k ≥ 2 let L r,k h = { χ ∈ C 0 (S k h ), χ = χ • a -1 k , for some χ ∈ L r h }. (4.4.5)
Analogously to (4.1.1), we dene

min u∈L r,k h ∇ S k h u ≤1 u(b)=0 F k h (u) (4.4.6)
where

F k h (u) = S k h ∇ S k h u 2
-mu and b some xed nodal points of the mesh T h .

Convergence of the lifted minimizers

In order to prove the convergence of our numerical approach, let us rst establish that our discrete problem converges in values in the sense of Proposition 4.4.2. For a function u dened on S k h , we introduce its lifted function u l onto S dened by the relation u l (b) = u(x) for b ∈ S where x is the unique point of S k h which satises a(x) = b. Below, we focus our analysis in the piecewise linear case r = k = 1 which contains all the main ingredients of a proof for the general (r, k) case. For every h > 0, the convex optimization problem (4.4.6) has a unique solution.

Lemma 4.4.1. The dierential of the projection a onto S, when restricted to the tangent space of S h , is the identity, up to order 2 in h:

Da | T S h = Id + O(h 2 ).
The second dierential of a, when restricted to the tangent space of S h , is null, up to order 1 in h:

D 2 a | T S h = O(h).
Proof. The identity estimate on Da is a direct consequence of [31, equations (4.12), (

= r = 1. Let Lu l m,h := u l m,h |∇ S u l m,h | L∞(S) be the 1-Lipschitz normalization of u l m,h . Then, Lu l m,h ∈ H 1 (S)
and

F (Lu l m,h ) = min u∈H 1 (S) |∇ S u|≤1 u(b)=0 F (u) + O(h 1 2 ).
Proof. step 1. Let u m be the solution of problem (4.1.1). For ε > 0, let u m,ε : S → R be dened by:

u m,ε = d b (x) 2 2ε if d b (x) ≤ ε u m (x) -ε 2 if d b (x) ≥ ε.
According to lemma 3.2.3, we have u m = d b in a neighborhood of b. Therefore, for ε > 0 small enough, we have u m = d b on B(b, 2ε). In particular, we deduce that u m,ε is C

1 on S. As d 2 b is smooth in a neighborhood of b, the gradient of d 2 b /2ε is O(ε -1 )-Lipschitz on B(b, ε). Moreover, as u m = d b on B(b, 2ε), the gradient of u m is O(ε -1 )-Lipschitz on B(b, 2ε) \ B(b, ε).
According to Proposition 3.2.4, u m is also locally C 1,1 on S \ {b}. Therefore its gradient is O(ε -1 )-Lipschitz on S \ B(b, ε). Putting ths facts together, we obtain that u m,ε is C 1,1 on S, and the Lipschitz constant of its gradient is O(ε -1 ). Furthermore, as d b and u m are both 1-Lipschitz,

we have |∇u m,ε | ≤ 1. Now for ε > 0, consider v h,ε := I h u m,ε |∇ S h I h u m,ε | L∞(S h )
,

where I h u m,ε is the P 1 Lagrange interpolation of u m,ε on S h . For x ∈ S h , observe that we have the relation

I h u m,ε (x) = I h (u m,ε • a)(x) which says that I h u m,ε is the standard (at)
interpolation of the composed function u m,ε • a. From lemma 4.4.1, we know that on every triangle, the dierential of a is O(h)-Lipschitz, and a is O(1)-Lipschitz. As the gradient of u m,ε is O(ε -1 )-Lipschitz, we deduce that on every triangle, the gradient of u m • a is O(ε -1 )-Lipschitz. By the quasi uniformity of the mesh, we obtain the uniform interpolation estimates on S h :

I h u m,ε (x) = (u m,ε • a)(x) + O(ε -1 h 2 ) (4.4.7)
and

∇ S h I h u m,ε (x) = ∇ S h (u m,ε • a)(x) + O(ε -1 h).
With lemma 4.4.1, we deduce for all x ∈ S h ,

∇ S h I h u m,ε (x) = ∇ S u m,ε (a(x)) + O(ε -1 h). (4.4.8)
Recall that we have |∇ S u m,ε | L∞(S) = 1. Therefore the last identity yields

|∇ S h I h u m,ε | L∞(S h,ε ) = 1 + O(ε -1 h). Thus, v h,ε = I h u m,ε (1 + O(ε -1 h))
, and so

F h (v h,ε ) = F h (I h u m,ε ) + O(ε -1 h). (4.4.9)
Applying lemma 4.4.1 again, with a simple change of variable, we nd that for any function

f : S h → R, S h f • a = S f + O(h 2 ).
Recalling (4.4.7) and (4.4.8), we obtain

F h (I h u m,ε ) = F (u m,ε ) + O(ε -1 h). (4.4.10)
Furthermore, we have

S |u m,ε -u m | ≤ O(ε) and S |∇u m,ε -∇u m | 2 ≤ O(ε 2 ), so F (u m,ε ) = F (u m ) + O(ε).
Combining this with (4.4.9) and (4.4.10), we nd

F h (v h,ε ) = F (u m ) + O(ε -1 h) + O(ε). Choosing ε = h 1 2 , this yields min u∈H 1 (S h ) ∇ S h u ≤1 u(b)=0 F h ≤ min u∈H 1 (S) |∇ S u|≤1 u(b)=0 F + O(h 1 2 ).
:= u m,h • (a | S h
) -1 its lifted version on S, and Lu l m,h :=

u l h |∇ S h u l h | L∞(S h )
. We show as before, using the

equation u m,h = u l h • a, that F (Lu l m,h ) = F h (u m,h ) + O(h). With (4.4.11), this implies min u∈H 1 (S) |∇ S u|≤1 u(b)=0 F ≤ F (Lu l m,h ) ≤ min u∈H 1 (S) |∇ S u|≤1 u(b)=0 F + O(h 1 2
), which concludes the proof of the proposition.

We can now establish the convergence of the minimizers: Proposition 4.4.3.

∇u l m,h -∇u m 2 L 2 (S) = O(h 1 2 ) and u l m,h -u m L 1 (S) = O(h 1 2 ). Proof. Consider v = 1 2 (Lu l m,h + u m ).
Then, v is admissible for problem (4.1.1), so F (v) ≥ F (u m ). Moreover, the following algebraic identity holds

F (v) = 1 2 F (Lu l m,h ) + 1 2 F (u m ) - 1 4 S |∇ S u m -∇ S Lu l m,h | 2 .
Therefore, we have

1 2 F (Lu l m,h ) - 1 2 F (u m ) ≥ 1 4 S |∇ S u m -∇ S Lu l m,h | 2 ,
which proves, with Proposition 4.4.2, that

∇Lu l m,h -∇u m 2 L 2 (S) = O(h 1 2 ).
F (Lu l m,h ) -F (u m ) = S ∇Lu l m,h 2 -|∇u m | 2 -m S Lu l m,h -u m .
The last two equations imply

Lu l m,h -u m L 1 (S) = O(h 1 2 ).
m,h = u l m,h • h, we show that Lu l m,h = u l m,h (1 + O(h 2 )
). Together with (4.4.12) and (4.4.13), this concludes the proof.

We just proved that the sequence of the lifted minimizers converges with an order at least 1/2 to the minimizer of problem (4.1.1). By analogy with the more standard variational context [START_REF] Demlow | Higher-order nite element methods and pointwise error estimates for elliptic problems on surfaces[END_REF][START_REF] Dziuk | Finite element methods for surface pdes[END_REF], we expect a convergence of order O((h r + h k+1 ) 1

2 ) using an approximation of orders (r, k).

Convergence in measure

Let us recall that the set E m,λ is dened by

E m,λ = x ∈ S \ {b} : |∇ S u m (x)| 2 ≤ 1 - λ 2 u 2 m (x)
. Proposition 4.4.4. For any λ > 0 and ε > 0 with ε < λ/2, let us dene

E m,λ,h := x ∈ S \ {b} : ∇ S u l m,h (x) 2 ≤ 1 - λ 2 (u l m,h ) 2 (x)
.

Then, we have

|E m,λ+ε \ E m,λ,h | = O(h 1 2 ) and |E m,λ,h \ E m,λ-ε | = O(h 1 2 ).
Proof. By denition of E m,λ and E m,λ,h , we have

E m,λ+ε \ E m,λ,h ⊂ ∇u l m,h 2 -|∇u m | 2 > (λ + ε) 2 u 2 m - λ 2 (u l m,h ) 2 .
Therefore, on E m,λ+ε \ E m,λ,h , we have

∇u l m,h 2 -|∇u m | 2 > (λ + ε) 2 -λ 2 u 2 m + λ 2 ( 1 u 2 m - 1 (u l m,h ) 2 ) ≥ 2ελ + ε 2 u 2 m -λ 2 2 min(u m , u l m,h ) 3 u m -u l m,h = 2ελ + ε 2 (diam S) 2 -λ 2 2 (u m + O(h 1 2 )) 3 u m -u l m,h ,
where diam S is the diameter of S. By denition of E m,λ , we also have

E m,λ+ε ⊂ {u m ≥ (λ + ε)}, so on E m,λ+ε \ E m,λ,h , ∇u l m,h 2 -|∇u m | 2 > 2ελ + ε 2 (diam S) 2 -λ 2 2 (λ + ε + O(h 1 2 )) 3 u m -u l m,h .
So for h large enough, we have

E m,λ+ε \ E m,λ,h ⊂ ∇u l m,h 2 -|∇u m | 2 + 2λ u m -u l m,h > 2ελ + ε 2 (diam S) 2 . ⊂ ∇u l m,h 2 -|∇u m | 2 > 2ελ + ε 2 2(diam S) 2 ∪ u m -u l m,h > 2ελ + ε 2 4λ(diam S) 2 .
Now from Proposition 4.4.3, we know that for any η > 0, we have the following estimates

η ∇u l m,h 2 -|∇u m | 2 > η ≤ S ∇u l m,h 2 -|∇u m | 2 = O(h 1 2 ),
and

η u m -u l m,h > η ≤ S u m -u l m,h = O(h 1 2
).

This gives the estimate |E m,λ+ε \ E m,λ,h | = O(h 1 2
). The other estimate is proved by the same method.

Remark 4.4.5. We expect a convergence of order O((h r + h k+1 ) 1

2 ) using an approximation of orders (r, k).

All the preceding sections together justify that the set E m,λ,h is a good approximation of the cut locus of b in M , if m is large enough, and λ and h are small enough.

4.5

Numerical illustrations

Cut locus approximation

We established the convergence of the minimizers of solutions of problems (4.1.1) when h tends to 0. For a xed h > 0, this convex discrete problems is of quadratic type with an innite number of conic pointwise constraints. By the way, it is important to observe that for k = r = 1, the gradient pointwise bounds for a function of P 1 is equivalent to a single discrete conic constraint on every triangle with respect to the degrees of freedom of P 1 (T h ). Nevertheless, we observed in our experiments that using P 1 elements may lead to approximated cut loci with some tiny articial connected components. Motivated by this lack of precision, we use in all following illustrations elements of order r > 1.

For the general case r > 1, the bound constraint on the gradient can not be easily reduced to a nite set of discrete constraints. In our experiments, we approximated the constraint

|∇ S k h u| L ∞ (S k h
) ≤ 1 by forcing this constraint only on a nite number of points of the mesh. In practice, we imposed these constraints on the Gauss quadrature points of order g on every triangle of T h . We illustrate in Figures 4.1, 4.2, 4.3 and 4.4 the approximation of the cut locus provided by our approach. These computations have been carried out on meshes of approximated 10 5 triangles for k = 2 and r = 3 using high precision quadrature formula associated to 17 Gauss points on every element of the mesh. Moreover, for r = 3, we imposed the conic gradient constraints on the g = 9 Gauss points of every triangle. In order to solve the resulting linear conic constrained quadratic optimization problem, we used the JuMP modeling language and the nite elements library Getfem++ [START_REF] Dunning | Jump: A modeling language for mathematical optimization[END_REF][START_REF] Renard | Getfem++. An open source generic C++ library for nite element methods[END_REF] combined with Mosek optimization solver [START_REF] Andersen | The mosek interior point optimizer for linear programming: an implementation of the homogeneous algorithm[END_REF].

For such a precision, the optimization solver identied a solution in less than one hour on a standard computer.

Approximation of the boundary of Voronoi cells

All previous theoretical results still hold if we replace the source point b by any compact subset of the surface S. For instance, if b is replaced by a set of points, the singular set of the distance function can be decomposed as the union of the boundary of voronoi cells and the cut loci of every point intersected with its voronoi cell. As a consequence, if the distribution of source points is homogeneous enough, that is every voronoi cell is small enough, the singular part of the distance function will be exactly equal to the boundary of the voronoi cells. We illustrate this remark in the following experiments. We used exactly the same framework as in previous sections and just replaced the pointwise condition at b with the analogous pointwise Dirichlet conditions at every source point. where the Γ l ij are the Christoel symbols of the metric g. As γ is unit speed, the ( γi ) are bounded, uniformly in γ. Therefore, there exists a constant α > 0 independent of γ such that |γ| ≤ α. By integration, we nd |γ(t) -γ(a) -γ(a

)(t -a)| ≤ α(t -a) 2 . Evaluating this expression at b yields |γ(b) -γ(a) -γ(a)(b -a)| ≤ α(b -a) 2 . From these two estimates, we deduce γ(t) -γ(a) - γ(b) -γ(a) b -a (t -a) ≤ α(t -a) 2 + α(b -a)(t -a). Taking t = (1 -λ)a + λb in this estimate yields |γ((1 -λ)a + λb) -((1 -λ)γ(a) + λγ(b))| ≤ αλ(1 + λ)(b -a) 2 = α(1 + λ) 1 -λ λ(1 -λ)(b -a) 2 .
Taking B := α(1+1/2) 1-1/2 , this proves the desired estimate when λ ≤ 1 2 . This concludes the proof.

We can now prove that dening local semiconcavity through charts (as in [START_REF] Mantegazza | HamiltonJacobi Equations and Distance Functions on Riemannian Manifolds[END_REF]), or through geodesics, is the same.

Proposition A.1.4. Let u : M → R be a locally Lipschitz function. Then, u is locally semiconcave if and only if for any chart ψ of M , u • ψ -1 is locally semiconcave as a function on R n . Proof. Let us assume that u is locally semiconcave. Let ψ : U → V be a chart from an open set U of M to on open set V of R n , and y ∈ V . Let f := u • ψ -1 . We want to show that f is semiconcave in a neighborhood of y, as a function of R n . We rst observe that f is locally semiconcave on the manifold (V, ψ g). Let V ⊂ V be a neighborhood of y that is geodesically convex for the metric ψ g, and such that there exists a constant C > 0 such that f is Csemiconcave on (V , ψ g). Let d denote the distance function on (V , ψ g). Up to taking V smaller, we may assume that the metric ψ g is bounded on V , and so there exists a constant

β > 0 such that ∀x, y ∈ V , d(x, y) ≤ β |x -y| . Let x, y ∈ V be such that [x, y] ⊂ V , and λ ∈ [0, 1]. Let γ : [a, b] → V be a unit speed geodesic of (V , ψ g) from x to y. By the C-semiconcavity of f on (V , ψ g), we have λ f (x)f (y) -f (λ xy ) = λ f (γ(a))f (γ(b)) -f (λ γ(a)γ(b) ) ≤ Cλ(1 -λ)(b -a) 2 + f (γ(λ ab )) -f (λ γ(a)γ(b) ) ≤ Cλ(1 -λ)(b -a) 2 + Lip(f ) γ(λ ab ) -λ γ(a)γ(b) .
Applying lemma A.1.3 above, we get a constant B > 0 such that

λ f (x)f (y) -f (λ xy ) ≤ (C + Lip(f )B)λ(1 -λ)(b -a) 2 = (C + Lip(f )B)λ(1 -λ)(d(x, y)) 2 ≤ (C + Lip(f )B)β 2 λ(1 -λ) |x -y| 2 , Proof. Let ε > 0. Let q ∈ M be such that |∇u ∞ (p ∞ )| ≤ u ∞ (q) -u ∞ (p ∞ ) d(q, p ∞ ) -ε. Then, we have |∇u ∞ (p ∞ )| ≤ lim inf k→∞ u k (q) -u k (p k ) d(q, p k ) -ε ≤ lim inf k→∞ |∇u k (p k )| + 2Cd(q, p k ) -ε = lim inf k→∞ |∇u k (p k )| -ε,
where we have used the Proposition A.1.7 above in the second estimate. As it is valid for any ε, this concludes the proof.

We recall the following proposition.

Proposition A.1.9 (Proposition 3.4 in [START_REF] Mantegazza | HamiltonJacobi Equations and Distance Functions on Riemannian Manifolds[END_REF].). For any compact subset K of a smooth manifold without boundary M , the distance function to the set K is locally semiconcave on M \ K.

We end this appendix with the following proposition about the generalized gradient of the distance function to a general compact set K.

Proposition A.1.10. Let M be a smooth Riemannian manifold without boundary, K a compact subset of M , and d K the distance function to K. Let p be a point of M such that there exist several minimizing geodesics from p to K. We denote the set of unit speed geodesics from p to K that are minimizing between p and K by geod(p, K). For any v ∈ T p M , we have

∂ + v d K (p) = min γ∈geod(p,K) -γ(0) • v.
In particular,

|∇d K | (p) = max{0, max v∈TpM,|v|=1 min γ∈geod(p,K) -γ(0) • v}.
A similar result for the distance function to a point is proved in [1, Theorem 4.5.6], in the more general framework of Alexandrov spaces, but with some additional restrictions on the curvature of the ambient space. A similar result for the directional derivatives of the function

d(•, •) is also stated in [1, Lemma 3.2]
, but the proof given there mainly consists in saying that the proof in [1, Theorem 4.5.6] also works in the case of Riemannian manifolds without the restriction on the curvature. For any reader that is not familiar with Alexandrov spaces, that is not very satisfying. So here is a complete proof in the Riemannian context.

Proof. Let γ : [0, d K (p)] → M be a geodesic of geod(p, K). Let a = γ(d K (p)/2). As γ is minimizing between p and γ(d K (p)), we have a / ∈ Cut p (M ), and so p / ∈ Cut a (M ). In particular, the function d a is dierentiable at p, and ∇d a (p) = -γ(0). Thus, we have

d K (exp p (tv)) -d K (p) t ≤ d a (exp p (tv)) + d K (a) -d K (p) t = d a (exp p (tv)) -d a (p) t → -γ(0) • v as t → 0.

A.2 Tensor calculus and covariant derivatives

In this appendix, we recall some notions of tensor calculus and covariant derivation on Riemannian manifolds. These notions are used in the proof of Proposition 3.5.5 in particular. If the reader has studied these notions before, we hope that these reminders will be sucient to understand the computations of the proof of Proposition 3.5.5. We try and give a denition of every notion needed, along with the computation rules we need. We give a list of formulae at the end of the section. If the reader is totally unfamiliar with these notions, those reminders might be too brief for an introduction.

Let us give some specic references for each notion. A brief introduction to the notions of tensor, tensor product, trace of a tensor, tensor eld over a manifold, can be found in [55, Chapter 3]]. For the notions of raising and lowering indices, inner product on tensors, see [55,

Chapter 3]]). For covariant derivatives, see [START_REF] Lee | Introduction to Riemannian Manifolds[END_REF]Chapter 4]).

Tensors. Let V be a n-dimensional vector space (you may think of V as the tangent space of a point p ∈ M , T p M ). Let k, l ∈ N. For any i ∈ {1, ..., k + l}, let V i denote either V or its dual V . A (k, l)-tensor on V is a multilinear map

T : V 1 × ... × V k+l → R,
where V appears k times in the product V 1 × .. × V k+l and V appears l times. In this case we say that T is a tensor of rank k + l.

In particular, a (0, 1) tensor is just a linear form on V , and a (1, 0) tensor is an element of V , so it can be seen as a vector of V . Reciprocally, any vector X ∈ V will automatically be seen as a (1, 0) tensor that will still be denoted by X. More generally a tensor T :

V × ... × V k × V × ... × V l → R can be seen as an element of the tensorial product V ⊗ ... ⊗ V k ⊗ V ⊗ ... ⊗ V l .
Let (e i ) 1≤i≤n be a basis of V , whose dual basis is (e i ) 1≤i≤n . The coordinates of a (k, l)

tensor T : V × ... × V k × V × ... × V l → R in
the basis (e i ) 1≤i≤n are dened by T j 1 ...j k i 1 ...i l := T (e i 1 , ..., e j k , e i 1 , ...e i l ).

Note that the indices to corresponding arguments in V are subscripts, while the indices corresponding to arguments in V are superscripts. If T were to take arguments in V and V in a dierent order, then we would write the indices in the coordinates ot T in the corresponding order, still writing subscripts for indices corresponding to arguments in V and superscripts for indices corresponding to arguments in V . For instance, if S :

V × V × V → R is a (1, 2)
tensor we will write S j i k := S(e i , e j , e k ).

Abstract index notation. We may call a tensor by its coordinates, i.e we may write 'the tensor S b a c ' instead of 'the tensor S'. This way we can encode the type of a tensor directly in its name. This will also prove useful for computations. However, when doing so, we will use letters a, b, c, d... while we keep the letters i, j, k, l... for actual coordinates in a basis. Therefore, T b 1 ...b k a 1 ...a l is non-other than the tensor T , while the T j 1 ...j k i 1 ...i l are the coordinates of the tensor T in the basis we have chosen. One may think of the a i and b i as mere labelling of the slots of T . This convention is called abstract index notation.

Tensor product. Given two tensors S and T of rank p and q respectively, their tensor product is a tensor of rank p + q dened by S ⊗ T (u 1 , ..., u p , v 1 , ...v q ) = S(u 1 , ...u p )T (v 1 , ...v q ).

The coordinates of S ⊗ T are the products of the coordinates of S and T . For instance, the coordinates of the tensor product of S b a c and T d e are

(S ⊗ T ) j l i k m = S j i k T l m .

Therefore, using the convention of abstract index notation, the tensor product of S b a c and T de will be denoted by S b a c T de .

Trace of a tensor. Let T b a be a (1, 1) tensor. For any X ∈ V , T (X, •) ∈ V V , so T can also be seen as an endomorphism of V . Its trace as a tensor, denoted by tr(T ) is dened as the trace of the associated endomorphism of V . The trace of T is also called contraction of T . In any basis (e i ) of V , we have the formula tr(T ) = n i=1 T i i .

We will use the Einstein summation convention, where an index repeated twice means we are summing over that index. Therefore, we have tr(T ) = T i i .

In abstract index notation, we will denote by T a a the trace of T . We can extend this denition to any (k + 1, l + 1) tensor T . All we need is a lower index and an upper index. The result is then a (k, l) tensor. However, one must specify with respect to which indices the trace is taken. This can be done when writing the coordinates of the tensor. For instance, we can take the trace of the tensor S cd ab with respect to the rst and last indices, to get a (1, 1) tensor whose coordinates are S ki ij . We will denote this tensor by S ca ab . The tensor S ba ab would denote the trace of the tensor S with respect to the rst and fourth coordinates, and the second and third coordinates, taken successively (the order does not matter.)

As mentioned before, any vector X of V can be seen as a (1, 0) tensor. We still denote by X the associated [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]0) tensor. Now let T a 1 ...a k b 1 ...b l+1 be a (k, l + 1) tensor. Then, T ( • , ..., • , X) is a (k, l) tensor, and one can see (using coordinates) that it is the trace of the tensor product of T and X, i.e

T ( • , ..., • , X) a 1 ...a k b 1 ...b l = T a 1 ...a k b 1 ...b l c X c .
The tensor T a 1 ...a k b 1 ...b l c X c is also called contraction of T and X. Therefore, the result of the action of a tensor on a vector is the contraction of this tensor with the vector (seen as a (1, 0) tensor). Likewise, the result of the action of any tensor T a 1 ...a k b 1 ...b l on the arguments φ 1 ,..., φ k , X 1 ,...,X l , is

T (φ 1 , ..., φ k , X 1 , ...X l ) = T a 1 ...a k b 1 ...b l φ 1 a 1 ...φ k a k X b 1 1 ...X b l l .
Raising and lowering indices. Now we assume that V is equipped with a scalar product g. (If V is T p M , g can be the metric of the manifold M .) The scalar product induces an isomorphism between V and its dual V . We will also denote this isomorphism by g. Its inverse is denoted by g -1 . By composition with this isomorphism, we can transform a (0, 1)

tensor T into a (1, 0) tensor T and vice-versa by setting T (X) = T (g -1 (X)).

We denote by g ij and g ij the components of the maps g and g -1 respectively. In coordinates, we have the relations

T i = g ij T j and T j = g ij T i
We will denote the tensor T by T a . We have

T a = g ab T b ,
which means that T a is the contraction of g ab and T b , or equivalently the trace of the tensorial product of g ab and T b . We call this operation 'raising' and index. We may also lower an index thanks to the relation

T a = g ab T a .
Likewise, we can transform any (k, l + 1) tensor into a (k + 1, l) tensor, and vice-versa, by composition with the isomorphism g, or equivalently by taking the trace of the tensorial product with g. We may also raise several indices successively. The order does not matter.

For instance, for a tensor S ab , we have S ab = g ac g bd S cd .

More about the trace. With the operation of raising and lowering indices, we can dene the trace of a tensor with respect to any two indices, even if they are both low or both up:

if this two indices are low, raise one of them and take the trace, otherwise lower one of them and take the trace. For instance, the trace of the tensor S ab is S a a (= S ab g ba = tr(S ab g bc )).

Inner product of tensors. We can extend the scalar product g on V to a scalar product on tensors of the same type, by requiring that if (e i ) is an orthonomal basis and (e i ) its dual, then the family composed of tensor products of elements of (e i ) and (e i ) of a given type is orthonormal. It can also be described in terms of raising and lowering indices and taking the trace. Indeed, to take the scalar product of two tensors T and S of the same type, raise all the lower indices and lower all the upper indices of T , and take the trace of the tensor product of T and S with respect to all pair of corresponding indices. For instance, for some tensors T ab and S ab , we have g(T, S) = T ab S ab = g ac g bd T cd S ab .

We also dene the norm of a tensor T by |T | 2 = g(T, T ).

(A.2.1)

T on X and Y is not the same as derivating T in the directions Y and X successively. Indeed, we have D Y T = D b T Y b , so some derivatives of Y b will appear in D X (D Y T ).

Laplacian. The Laplacian of a function u is the trace of the second covariant derivative of u,

i.e.

∆u := g ab D a D b u.

Riemann curvature tensor. The Riemann curvature tensor is a (0, 4

) tensor R abcd dened by ∀X, Y, Z, W ∈ X(M ), R(X, Y, Z, W ) = g(D 2 Z(X, Y ) -D 2 Z(Y, X), W ).
Though Z needs to be a vector eld (and not just a vector) for this denition to make sense a priori, it only depends on the pointwise values of Z, and so R is actually a tensor. We will denote the commutator of D a and D b by D 2

[ab] :

D 2 [ab] := D a D b -D b D a .
With this notation, we have for any vector Z:

D 2 [ab] Z c = R c abd Z d .
Summary of useful formulae. Here we specically recall the formulas we use in the proof of Proposition 3.5.5, in abstract index notation.

• Trace: we recall that when two (abstract) indices are repeated, it means we are taking the trace with respect to those indices.

• Tensor applied to vectors:

T (X, Y ) = T ab X a Y b .
• Norm of a tensor: T ).

• Compatibility with the metric: symbols D and g can be inverted.

P (E) + V α (E) + U β,A (E), (1.1.3)
where

• P (E) is the perimeter of E,

• V α (E) := E×E dxdy |x-y| n-α , • U β,A (E) := E A |x| β dx,
• n ∈ N * (the dimension of the ambient space), m > 0 (called the mass), α ∈ (0, n) and A ≥ 0 are constants.

If A = 0, we recover Gamow's liquid drop model, which does not admit any minimizer for m suciently large. If A > 0, this problem admits a minimizer for any mass m.

In the rst chapter, we study minimizers of large mass (i.e when m → +∞). We show the following.

Theorem (Theorem 1.1.1). Given α ∈ (0, n), β > 0 and A > 0, assume α < β. Let (E m ) m>0 be a family of minimizers in However, from Proposition 1.3.3 we know that if β < 1, then large volume minimizers are not exactly balls. Indeed, in this case large balls are not even locally minimizing. On the contrary, in the case β > 1, if we assume in addition that α > 1, then we have: Theorem (Theorem 1.1.2). Given α ∈ (0, n), β > 0 and A > 0, assume 1 < α < β. There exists a mass m 1 = m 1 (n, α, β, A) > 0 such that for any m > m 1 the ball of volume m centered at 0 is the unique minimizer (1.1.3), up to a set of measure 0.

In the second chapter, we develop a numerical method to compute the minimizers. Our results seem to conrm that the following conjecture of R. Choksi and M. Peletier regarding Gamow's model is true in dimension 2: for A = 0, if problem (1.1.3) admits a minimizer, then it is a ball.

We started working on some aspects of the rst Chapter under the supervision of Vincent Millot for our Master 2 thesis. The new results start with Section 1.4. In particular, the results presented in this introduction are new. Introduction

The classical isoperimetric problem

Let n ∈ N * be an integer. For any measurable set E ⊂ R n , the perimeter of E is dened as the total variation of the indicator function of E:

P (E) := R n |D1 E | = sup E div f : f ∈ C 1 c (R n ), |f | L ∞ ≤ 1 .
Our book of reference for the perimeter is Giusti's book, [START_REF] Giusti | Minimal surfaces and functions of bounded variation[END_REF]. Maggi's book [START_REF] Maggi | Sets of nite perimeter and geometric variational problems: an introduction to Geometric Measure Theory[END_REF] is probably more reader-friendly though (but does not include the study of BV functions). The quantity P (E) may be innite, and when it is nite, we say that E is a set of nite perimeter. If E is a C 1 open set, then as one would expect, we have P (E) = H n-1 (∂E), where H n-1 is the (n -1)-dimensional Hausdor measure. The properties of the perimeter we need will be recalled along the way, when needed. The classical isoperimetric problem is the following variational problem:

min E⊂R n ,|E|=m P (E),
where |E| is the volume of E, and m > 0 is a constant. It is well known that for any m > 0, this problem has a unique minimizer up to translation, namely B[m], the ball of volume m.

Furthermore, a sharp quantitative estimate was proved in [START_REF] Fusco | The Sharp Quantitative Isoperimetric Inequality[END_REF]. Before stating it, we need a denition. Given a set E of volume m, we call Fraenkel asymmetry the quantity λ(E) := min

x∈R n |E∆(x + B[m])| |E| ,
where E∆F = (E ∪ F ) \ (E ∩ F ) is the symmetric dierence of E and F . There exists a constant C(n) depending only on the dimension of the ambient space such that for any set E of volume m, P (E) -

P (B[m]) ≥ C(n)P (B[m])λ(E) 2 .
See also [START_REF] Fusco | A strong form of the quantitative isoperimetric inequality[END_REF] for a renement of this inequality, where the dierence P (E) -P (B[m]) also controls the oscillation of the normal to the (reduced) boundary of E. • m, called the mass, is a positive constant.

In this model, the nucleus is thought of as an incompressible charged liquid. The perimeter term models the short-range attractive nuclear force, whereas the other term, similar to an electrostatic energy, models the repulsive forces between the protons of the nucleus. Gamow invented this model in the 1930's, in an attempt to predict the shape of nuclei, the nonexistence of large nuclei, and the existence of a nucleus with the least energy per nucleon. These three goals have only been attained relatively recently (in the 2010's, see next subsection).

Problem (1.1.2) has been studied by mathematicians as an interesting extension of the classical isoperimetric problem. Indeed, two terms are competing: the perimeter tends to round things up and is minimized by balls, whereas the non-local term tends to spread the mass, and is maximized by balls. This last fact is a consequence of Riesz rearrangement inequalities (see [START_REF] Lieb | of graduate studies in mathematics[END_REF]).

Some variations of the liquid drop model

The most studied variant of Gamow's model is the following:

inf E⊂R 2 ,|E|=m P (E) + V α (E) (1.1.2)
where

• P (E) is the perimeter of E, • V α (E) := E×E dxdy |x-y| n-α is the Riesz energy of E,
• n ≥ 2 (the dimension of the ambient space), m > 0 and α ∈ (0, n) are constants.

The physical case corresponds to n = 3 and α = 2. In this case, the Riesz energy boils down to the electrostatic energy E×E dxdy |x-y| .

It was shown in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] by Figalli, Fusco, Maggi, Millot, and Morini, that if the mass m is small enough, then the problem (1.1.2) admits a unique minimizer (up to translation), namely the ball of volume m. See also the papers of Knüpfer, Muratov, Julin, Bonacini and Cristoferi [START_REF] Knüpfer | On an isoperimetric problem with a competing nonlocal term i: The planar case[END_REF], [START_REF] Muratov | On an isoperimetric problem with a competing nonlocal term ii: The general case[END_REF], [START_REF] Julin | Isoperimetric problem with a Coulombic repulsive term[END_REF] and [START_REF] Bonacini | Local and global minimality results for a nonlocal isoperimetric problem on R N[END_REF], for anterior and slightly less general results.

On the other hand, for α ∈ (n -2, n), it was shown in [START_REF] Muratov | On an isoperimetric problem with a competing nonlocal term ii: The general case[END_REF] that for m large enough there is no minimizer of problem (1.1.2). This result was simultaneously proved in [START_REF] Lu | Nonexistence of a minimizer for thomasfermidiracvon weizsäcker model[END_REF] by Lu and Otto in the physical case. See also [START_REF] Frank | Nonexistence of large nuclei in the liquid drop model[END_REF] for a short proof with a quantitative bound, by Frank, Killip and Nam.

In [START_REF] Frank | A Compactness Lemma and Its Application to the Existence of Minimizers for the Liquid Drop Model[END_REF], Frank showed in the physical case that there exists a set which minimizes the quantity (P (E) + V α (E))/ |E|, thus proving that in Gamow's model, there is a nucleus with the least energy per nucleon.

In [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF], the authors studied a generalization of problem (1.1.2), where the perimeter term is replaced with a non-local perimeter.

Another interesting variation consists in replacing the repulsive Riesz energy with a repulsive energy involving another kernel.

Let K ∈ L 1 (R n ). Problem (
P (E) + E×E K(x -y)dxdy,
with K(x) = |x| α-n . In [START_REF] Rigot | Ensembles quasi-minimaux avec contrainte de volume et rectiabilité uniforme. (Quasi-minimal sets with a volume constraint and uniform rectiability)[END_REF], Rigot considered the above problem with K an L 1 compactly supported function, and proved that it has a minimizer for any mass m. The following papers are posterior to the present work. In [START_REF] Pegon | Large mass minimizers for isoperimetric problems with integrable nonlocal potentials[END_REF], Pegon proved the same result in the case of radially symmetric non-increasing kernels K with some regularity and integrability conditions at innity. He also showed that, under the same conditions, after rescaling, the problem Γ-converges to the classical isoperimetric problem as the mass m goes to +∞. In [START_REF] Muratov | A Nonlocal Isoperimetric Problem with Dipolar Repulsion[END_REF], Muratov and Simon considered the kernel |x| -3 in dimension 2, and also studied the minimizers of large mass.

In [START_REF] Alama | Droplet breakup in the liquid drop model with background potential[END_REF], Alama, Bronsard, Choksi, and Topaloglu studied problem (1.1.2) with the addition of a conning background potential of the form -Z |x| p . They proved that it admits a minimizer for any mass m, and study the minimizers as Z → 0.

In [START_REF] Goldman | Quantitative estimates for bending energies and applications to non-local variational problems[END_REF], Goldman, Novaga and Roger studied the small mass minimizers of problem (1.1.2)

with the addition of the Willmore energy.

As shown in [START_REF] Choksi | Small volume-fraction limit of the diblock copolymer problem: Ii. diuse-interface functional[END_REF], Gamow's liquid drop model is also related to diblock copolymers, and the OhtaKawasaki functional. This is also an active eld of research, see for instance [START_REF] Alama | Ground-states for the liquid drop and tfdw models with long-range attraction[END_REF] and [START_REF] Van Den Berg | Optimal periodic structures with general space group symmetries in the Ohta-Kawasaki problem[END_REF].

Our problem

To restore the existence of a minimizer for large masses in (1.1.2), we add the energy associated to the potential A |x| β to our functional, as we expect it to counter the spreading eect of the V α term. Thus we are interested in the following modication of the original problem (1.1.2):

inf

E⊂R n ,|E|=m E α,β,A (E) := P (E) + V α (E) + U β,A (E), (1.1.3)
where

• U β,A (E) := E A |x| β dx,
• A ≥ 0 and β > 0 are constants.

As easily proved in Section 1.2, we indeed have the existence of a minimizer in (1.1.3) for any mass m.

In section 1.3, we extend some known results about minimality of small balls, and the domain (of masses m) of local minimality of balls (depending on the parameters α and β). We do not give complete proofs, but recall briey the techniques used in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] to get these results. In section 1.4, we study large volume minimizers (i.e. when m is large) of (1.1.3) when α < β. More precisely we prove the following theorems: Theorem 1.1.1. Given α ∈ (0, n), β > 0 and A > 0, assume α < β. Let (E m ) m>0 be a family of minimizers in (1.1.3), such that |E m | = m, and let E * m be the rescaling of E m of the same mass as the unit ball centered at the origin, B (ie

E * m = |B| m 1 n E m ).
Then, up to modifying the sets (E m ) by some sets of measure 0, the boundaries of the sets (E * m ) Hausdor-converge to the boundary of B as m → +∞.

However, from Proposition 1.3.3 we know that if β < 1, then large volume minimizers are not exactly balls. Indeed, in this case large balls are not even locally minimizing. On the contrary, in the case β > 1, if we assume in addition that α > 1, then we have: Theorem 1.1.2. Given α ∈ (0, n), β > 0 and A > 0, assume 1 < α < β. There exists a mass m 1 = m 1 (n, α, β, A) > 0 such that for any m > m 1 the ball of volume m centered at 0 is the unique minimizer (1.1.3), up to a set of measure 0.

We conjecture that this last theorem holds also for any α ∈ (0, 1] as long as β > 1.

However, this seems to require dierent techniques.

1.2

Existence of a minimizer in (1.1.3) In this section, we prove the following easy proposition:

E α,β,A (E k ) ≤ E α,β,A (B[m]
), for all k ∈ N.

(

P (E k ∩ B R j ) ≤ E α,β,A (B[m]
) for all k.

Therefore we can extract a L 1 -converging subsequence of (E k ∩ B R j ) k∈N . Doing that for all j ∈ N and using a diagonal argument, we get a subsequence of (E k ) k∈N that converges locally in L 1 to a Borel set E ⊂ R n . Using the lower semi-continuity of the perimeter and Fatou's lemma in V α (E k ) and U β,A (E k ), we get that

E α,β,A (E) ≤ lim inf k→∞ E α,β,A (E k ). (1.2.2) Now we show that |E| = m. By Fatou's lemma, from |E k | = m, we get |E| ≤ m. Also, for any j ∈ N, from the inequalities U β,A (E k \ B R j ) ≤ U β,A (E k ), U β,A ≤ E α,β,A and (1.2.1) we get U β,A (E k \ B R j ) ≤ E α,β,A (B[m]). (1.2.3) But U β,A (E k \ B R j ) = E k \B R j g(x)dx ≥ A j E k \ B R j . (1.2.4) Thus (1.2.3) and (1.2.4) together give E k ∩ B R j = m -E k \ B R j ≥ m - E α,β,A B[m] A j .
Taking the limit k → ∞, then j → ∞, we obtain

E ∩ B R j ≥ m - E α,β,A B[m] A j , then |E| ≥ m.

Extension of some known results

In this section we recall two known results about the variational problem (1.1.2), and extend it to (1.1.3), recalling only the techniques used in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF]. The rst result states that if the mass m is small enough, then problem (1.1.2) admits a unique (up to translation) minimizer, namely the ball of volume m. The same holds for problem (1.1.3):

Proposition 1.3.1. Given α ∈ (0, n), β > 0, A > 0, there exists a constant m 0 (n, α, β, A) > 0 such that for any m ∈ (0, m 0 ), problem (1.1.3) admits the ball of volume m centered at 0 as its unique minimizer. is not a L 1 -local minimizer in (1.1.2). As stated in the next theorem, the addition of the U β,A term may modify this situation, but we can still apply the techniques used in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] to get a similar result.

• λ k = k(n + k -2) is the k-th eigenvalue of the Laplacian on the sphere ∂B,

• µ α k is the k-th eigenvalue of the operator R α dened by

R α u(x) := 2 ∂B u(x) -u(y) |x -y| n-α dH n-1 (y), ∀u ∈ C 1 (∂B).
From there we deduce that, dening

S * = {γ > 0 : 1 -γ 1+α µ α k -µ α 1 λ k -λ 1 + γ 1+β Aβ λ k -λ 1 ≥ 0, ∀k ≥ 2}, (1.3.1)
if γ / ∈ S * , then there exists a vector eld X such that

φ X t (B) = |B| and E α,β,A,γ (φ X t (B)) < E α,β,A,γ (B) for t small enough. Thus B is not a L 1 -local minimizer of E α,β,A,γ if γ / ∈ S * . Now let us set S * = {γ > 0 : 1 -γ 1+α µ α k -µ α 1 λ k -λ 1 + γ 1+β Aβ λ k -λ 1 > 0, ∀k ≥ 2}. (1.3.2)
We assume γ ∈ S * and explain how to show that B is a L 1 -local minimizer of E α,β,A,γ . First, we note that it is true in a certain class of nearly spherical sets. More precisely, let E be a nearly spherical set associated to a C 1 function u : ∂B → R:

E := {s(1 + u(x))x, x ∈ ∂B, 0 ≤ s ≤ 1}.
Assume that |E| = |B| and E |x| β-2 xdx = 0. Then, using some explicit computations and Taylor expansions, we can show that there exist some constants ε(n, α, β, A, γ) > 0 and C

(n, α, β, A, γ) > 0 such that if u C 1 (∂B) ≤ ε(n, α, β, A, γ), then E α,β,A,γ (E) -E α,β,A,γ (B) ≥ C(n, α, β, A, γ) u 2 L 2 (∂B) + ∇(u) 2 L 2 (∂B) . (1.3.3)
Next we argue by contradiction and assume that we have a sequence of Borel sets (E k ) such that for any k,

|E k | = |B|, E α,β,A,γ (E k ) < E α,β,A,γ (B[m]) and |E k ∆B| -→ k→∞ 0. We consider a set F k solution of the penalized problem: inf{E α,β,A,γ (E) + M |E∆E k | , E ⊂ R n },
with M > 0 to be taken large enough. The role of the set F k is to be "close to E k ", and to be a Λ-minimizer in the sense that

P (F k ) ≤ P (E) + Λ |E∆F k | ,
γ → 1 -γ 1+α µ α k -µ α 1 λ k -λ 1 + γ 1+β Aβ λ k -λ 1 , k ≥ 2
to get the conclusions of the theorem. This is done in Appendix 1.A.

1.4

Large volume minimizers for α < β The idea behind the proof is that if α < β, then for a Borel set E ⊂ R n of volume m > 0 with m large, the predominant term in E α,β,A (E) is U β,A (E). This can be seen by rescaling:

E α,β,A (E) = γ n-1 P (E * ) + γ 1+α V α (E * ) + γ 1+β U β,A (E * ) ,
= P + γ 1+α V α + γ 1+β U β,A , (1.4.2)
then, up to modifying each set E γ by a set of measure 0, the boundaries of the sets (E γ ) Hausdor-converge to the boundary of the unit ball B centered at 0 as γ → +∞. First we will show the following convergence in measure: Lemma 1.4.1. We have

|E γ ∆B| -→ γ→∞ 0.
We will need the following stability lemma for the potential energy U β,A . Lemma 1.4.2. For any Borel set E ⊂ R n of volume |B|, we have

U β,A (E) -U β,A (B)) ≥ Aβ 8P (B) |E∆B| 2 .
Proof. Let E ⊂ R n be a Borel set of volume |B|. Dene r 1 ≥ 0 and r 2 > 0 to be such

that |{x ∈ R n : r 1 ≤ |x| ≤ 1}| = |{x ∈ R n : 1 ≤ |x| ≤ r 2 }| = |E \ B| = |B \ E|. Explicitely, r 1 = 1 -n |E\B| P (B) 1 n and r 2 = 1 + n |E\B| P (B) 1 n
. We then have

U β,A (E) -U β,A (B) = E\B A |x| β dx - B\E A |x| β dx ≥ {x∈R n : 1≤|x|≤r 2 } A |x| β dx - {x∈R n : r 1 ≤|x|≤1} A |x| β dx (for x → |x| β is radially symmetric non-decreasing), = AP (B) r 2 1 r β r n-1 dr - 1 r 1 r β r n-1 dr = AP (B) n + β r n+β 2 -1 -1 -r n+β 1 = AP (B) n + β 1 + n |E \ B| P (B) n+β n -1 -1 -1 -n |E \ B| P (B) n+β n . (1.4.3)
Now, setting λ := n+β n and f (r

) := (1 + r) λ -1 -1 -(1 -r) λ , we have f (r) = λ (λ -1) (1 + r) λ-2 + (1 -r) λ-2 ≥ λ (λ -1)
As f (0) = f (0) = 0, we get f (r) ≥ λ(λ -1) r 2 2 , which yields the result with (1.4.3).

Lemma 1.4.1 is then easily deduced from Lemma 1.4.2 :

Proof of Lemma 1.4.1. We have

γ 1+β U β,A (E γ ) ≤ E α,β,A,γ (E γ ) ≤ E α,β,A,γ (B) = P (B) + γ 1+α V α (B) + γ 1+β U β,A (B), so U β,A (E γ ) -U β,A (B) ≤ 1 γ 1+β P (B) + γ 1+α V α (B) . This implies U β,A (E γ )-U β,A (
= P + γ 1+α V α + γ 1+β U β,A .
We need to show that, up to modifying each set E γ by a set of measure 0, the boundaries of the sets (E γ ) Hausdor-converge to the boundary of the unit ball B centered at 0 as γ → +∞.

Step one. We show that given R > 1, for γ large enough we have E γ ⊂ B R , up to a set of measure 0.

Given R > 1, set F = µ (E γ ∩ B R ), with µ > 0 such that |F | = |B|, ie µ = |Eγ | |Eγ ∩B R | 1 n = 1 1-u 1 n , where u = |Eγ \B R | |Eγ | . We have E α,β,A,γ (F ) = µ n-1 P (E γ ∩ B R ) + µ n+α γ 1+α V α (E γ ∩ B R ) + µ n+β γ 1+β U β,A (E γ ∩ B R ) ≤ µ n+β E α,β,A,γ (E γ ∩ B R ).
(1.4.4)

Take η > 0 to be chosen later, and then ε > 0 such that for all v ∈ [0, ε),

1 1-v n+β n ≤ 1 + n+β n + η v.
(E γ ∩B R ) ≤ P (E γ ), V α (E γ ∩B R ) ≤ V α (E γ ) and U β,A (E γ ) -U β,A (E γ ∩ B R ) ≥ A |E γ \ B R | R β , we nd E α,β,A,γ (F ) ≤ 1 + n + β n + η u E α,β,A,γ (E γ ∩ B R ) ≤ E α,β,A,γ (E γ ) -γ 1+β A |E γ \ B R | R β + n + β n + η uE α,β,A,γ (E γ ) = E α,β,A,γ (E γ ) + n + β n + η E α,β,A,γ (E γ ) -γ 1+β AR β |B| u ≤ E α,β,A,γ (E γ ) + n + β n + η E α,β,A,γ (B) -γ 1+β AR β |B| u. (1.4.5) But as γ → ∞, E α,β,A,γ (B) = γ 1+β U β,A (B) + o(γ 1+β ) = γ 1+β A 1 n + β P (B) + o(γ 1+β ) = γ 1+β A n n + β |B| + o(γ 1+β ). So with (1.4.5), E α,β,A,γ (F ) ≤ E α,β,A,γ (E γ ) + γ 1+β A |B| (1 + n n + β η -R β ) + o(γ 1+β ) u.
Step two. We show that given δ > 0, for γ large enough we have B 1-δ ⊂ E γ , up to a set of measure 0. This is done by taking some mass of E γ outside a certain ball B R and putting it in E γ ∩ B r for a well chosen r. In the proof we use Lemma 1.4.4 below to show that the perimeter decreases under such a transformation for a well chosen r ∈ (1 -δ, 1). On the other hand, the increase of V α is compensated by the decrease of U β,A if γ has been taken large enough.

Let us set F = B \ E γ and ε = δ/2.

= 1 -ε, to get a r ∈ (1 -δ, 1 -ε) such that P (F, B r ) ≥ H n-1 (F ∩ ∂B r ). (1.4.6) As |E γ | = |B|, we have |E γ \ B| = |B \ E γ | ≥ |B r \ E γ |, so there exists R ≥ 1 such that |E γ \ B R | = |B r \ E γ |. Now let us set E γ = (E γ ∩ B R ) ∪ B r ,
and compare E α,β,A,γ (E γ ) and E α,β,A,γ (E γ ). Using classical formulae for the perimeter of the union or the intersection of a set with a ball (see [43, remark 2.14]), we have

P (E γ ) = H n-1 (E γ ∩ ∂B R ) + P (E γ , B r c ∩ B R ) + H n-1 (E c γ ∩ ∂B r ), P (E γ ) = P (E γ , B c R ) + P (E γ , B r c ∩ B R ) + P (E γ , B r ), so that P (E γ ) -P (E γ ) = H n-1 (E γ ∩ ∂B R ) -P (E γ , B c R ) + H n-1 (E c γ ∩ ∂B r ) -P (E γ , B r ). (1.4.7)
From the classical inequality P

(E γ ∩B R ) ≤ P (E γ ), we get that H n-1 (E γ ∩∂B R ) ≤ P (E γ , B c R ), so (1.4.7) gives P (E γ ) -P (E γ ) ≤ H n-1 (E c γ ∩ ∂B r ) -P (E γ , B r ). But H n-1 (E c γ ∩ ∂B r ) = H n-1 (E c γ ∩ B ∩ ∂B r ) = H n-1 ((B \ E γ ) ∩ ∂B r ), and
P (E γ , B r ) = P (E c γ , B r ) = P (E c γ ∩ B, B r ) = P (B \ E γ , B r ), So, recalling that F = B \ E γ , we obtain P (E γ ) -P (E γ ) ≤ H n-1 (F ∩ ∂B r ) -P (F, B r ) ≤ H n-1 (F ∩ ∂B r ) -P (F, B r ),
so by the choice of r, P (E γ ) ≤ P (E γ ).

(1.4.8)

Now we estimate the variation of V α . Let us dene the non-local potential:

Φ α E (x) = E dy |x -y| n-α . 111 1.

Large volume minimizers for α < β

With this notation, we have

V α (E γ ) -V α (E γ ) = E γ Φ α E γ - Eγ Φ α Eγ = E γ Φ α E γ - Eγ Φ α E γ + E γ Φ α Eγ - Eγ Φ α Eγ = E γ \Eγ Φ α E γ - Eγ \E γ Φ α E γ + E γ \Eγ Φ α Eγ - Eγ \E γ Φ α Eγ ≤ 4 sup |F |=|B| Φ α F ∞ E γ \ E γ .
(1.4.9)

By the Hardy-Littlewood rearrangement inequality, we have, for any

F ⊂ R n such that |F | = |B| and x ∈ R n , Φ α F (x) = F dy |x -y| n-α = R n 1 {-x+F } |y| n-α dy ≤ R n 1 B |y| n-α dy = C(n, α). So (1.4.9) yields V α (E γ ) -V α (E γ ) ≤ C(n, α) |E γ \ B r | . (1.4.10) As for U β,A , we have U β,A (E γ ) -U β,A (E γ ) = Br\Eγ A |x| β dx - Eγ \B R A |x| β dx ≤ Br\Eγ Ar β dx - Eγ \B R AR β dx ≤ Br\Eγ A(1 -ε) β dx - Eγ \B R Adx = A (1 -ε) β |B r \ E γ | -|E γ \ B R | = A (1 -ε) β -1 |E γ \ B r | .
E α,β,A,γ (E γ ) -E α,β,A,γ (E γ ) ≤ γ 1+α C(n, α) + γ 1+β A (1 -ε) β -1 |E γ \ B r | .
As E γ is a minimizer, we have E α,β,A,γ (E γ )-E α,β,A,γ (E γ ) ≥ 0, so for α < β and γ large enough (depending only of n, α, β, A, δ), this last inequality implies |B r \ E γ | = 0, i.e. B r ⊂ E γ , up to a set of measure 0. This concludes Step two.

Step three. The rst and second steps show that for any k ∈ N * , there exists γ k > 0 such that for any γ > γ k , there exist some sets P γ,k ⊂ B and Q γ,k ⊂ B c of measure 0, such that

B 1-1/k ⊂ (E γ ∪ P γ,k ) \ Q γ,k ⊂ B 1+1/k .
For any γ > 0, let us dene the sets

P γ := {k∈N * :γ>γ k } P γ,k and Q γ := {k∈N * :γ>γ k } Q γ,k ,
and

E γ = (E γ ∪ P γ ) \ Q γ .
Then, for any γ > 0, E γ and E γ are equal up to a set of measure 0, and for any k ≥ 1, for any

γ > γ k , B 1-1/k ⊂ E γ ⊂ B 1+1/k .
In particular, we have (1.4.11)

Reciprocally, let y ∈ ∂B. For any k ≥ 1, for any γ > γ k , we have ).

(1 -2/k)y ∈ E γ and (1 + 1/k)y / ∈ E γ , so there exists t ∈ [1 -2/k, 1 + 1/k] such that ty ∈ ∂ E γ . In particular, d(y, ∂ E γ ) ≤ 2/k.
Lemma 1.4.4. Given F ⊂ R n a set of nite perimeter, r 0 > 0, and ε > 0, assume that

|F | ≤ |B| ε 2 n .
(1.4.12)

Then, there exists r ∈ (r 0 -ε, r 0 ) such that,

P (F, B r ) ≥ H n-1 (F ∩ ∂B r ).
(1.4.13)

Proof. We argue by contradiction and assume that (1.4.12) holds, and ∀r ∈ (r 0 -ε, r 0 ), P (F, B r ) < H n-1 (F ∩ ∂B r ).

Adding H n-1 (F ∩ ∂B r ) to both sides, this is equivalent to

P (F ∩ B r ) < 2H n-1 (F ∩ ∂B r ).
Using the isoperimetric inequality we get

|F ∩ B r | |B| n-1 n P (B) < 2H n-1 (F ∩ ∂B r ).
(1.4.14)

Now set for r ≥ 0, f (r) = |F ∩ B r |. We can assume f (r) = 0 for all r ∈ (r 0 -ε, r 0 ) otherwise the lemma is trivially true. We have for almost all r ∈ (0, ∞),

f (r) = H n-1 (F ∩ ∂B r ).
Thus (1.4.14) gives for almost all r ∈ (r 0 -ε, r 0 ),

1 n f (r)f (r) 1 n -1 > P (B) 2n |B| n-1 n = |B| 1 n 2 .
Integrating on the interval (r 0 -ε, r 0 ), we get

f (r 0 ) 1 n -f (r 0 -ε) 1 n > ε |B| 1 n 2 , so f (r 0 ) 1 n > ε |B| 1 n 2 ,
which contradicts (1.4.12).

1.4.2 Large volume minimizers = balls for α < β and β > 1

Here we prove Theorem 1.1.2, i.e. that if we assume in addition that α > 1, then large volume minimizers are exactly balls. We conjecture that the theorem is also true when α ∈ (0, 1], as long as β > 1. For β < 1, it cannot be true as we know from Proposition 1.3.3 that for m large the ball B[m] is not even a local minimizer. Note that in dimension 1, using Theorem 1.1.1, one can perform some computations to show that the theorem is indeed true under the more general assumption β > max(1, α).

The proof relies heavily on the following simple lemma:

Lemma 1.4.5. If α > 1, then there exists a constant C(n, α) > 0 such that for any set E ⊂ R n of volume |E| = |B|, we have

Φ α E C 1 (R n ) ≤ C(n, α), where Φ α E (x) = E dx |x -y| n-α .
This lemma is not true as soon as α ≤ 1, where we just get α-Hölder continuity instead of Lipschitz continuity. We refer to [10, Proposition 2.1] for a proof.

Proof of Theorem 1.1.2. Rescaling the functional as usual, we need to show that for γ large enough, if E ⊂ R n is such that |E| = |B|, and E is a volume-constrained minimizer of E α,β,A,γ (see (1.4.2)), then E=B. Let us show that for γ > 0 large enough, we have γ 1+α V α (E) + γ 1+β U β,A (E) ≥ γ 1+α V α (B) + γ 1+β U β,A (B). t n-1 dt, which gives a contradiction. This concludes the proof.

For each x ∈ ∂B, we apply this lemma to S = E x , to get a corresponding map T x . Then, Ar β -AT x (r) β r n-1 dr dH n-1 (x).

(1.4.23)

Now let us compute the variation of the Riesz energy V α in a similar fashion :

V α (E u ) -V α (E) = Eu Φ α Eu - E Φ α E = Eu Φ α Eu + Eu Φ α E - E Φ α Eu - E Φ α E = Eu (Φ α Eu + Φ α E ) - E (Φ α Eu + Φ α E ) = ∂B (1/2,u(x)) (Φ α Eu + Φ α E )(rx)
-(Φ α Eu + Φ α E )(T x (r)x) r n-1 drdH n-1 (x). From this inequality we get that if γ is large enough (depending only on n, α, β, A), then

γ 1+α V α (E u ) + γ 1+β U β,A (E u ) ≤ γ 1+α V α (E) + γ 1+β U β,A (E).
(1.4.25)

Step two. We show that there exists ε = ε(n, α, β, A) > 0, such that for any γ large enough, if u L ∞ (∂B) < ε, then γ 1+α V α (B) + γ 1+β U β,A (B) ≤ γ 1+α V α (E u ) + γ 1+β U β,A (E u ).

( |x -y| n-α dH n-1 (x)dH n-1 (y).

Indeed for α > 1, we have This implies that for t small enough, depending only on n and α, we have V α (E t ) ≥ V α (B) -t 2 C(n, α) u 2 L 2 (∂B) .

Likewise, we get from (1.4.27) that for t small enough, depending only on n, β and A, we have U β,A (E t ) ≥ U β,A (B) + t 2 C(n, β, A) u 2 L 2 (∂B) .

These last two inequalities imply that there exists ε = ε(n, α, β, A) > 0, such that if u L ∞ (∂B) < ε, then As stated in [33, equations (7.4), (7.5) and (7.6)], we have

γ 1+α V α (B) + γ 1+β U β,A (B) ≤ γ 1+α V α (E u ) + γ 1+β
µ α k =                2 1+α π n-1 2 1-α Γ( 1+α 2 ) Γ( n-α 2 ) Γ(k+ n-α 2 ) Γ(k+ n-2+α 2 ) - Γ( n-α 2 ) Γ( n-2+α 2 ) if α ∈ (0, 1), 2 α π n-1 2 Γ( α-1 2 ) Γ( n-α 2 ) Γ( n-α 2 ) Γ( n-2+α 2 ) - Γ(k+ n-α 2 ) Γ(k+ n-2+α 2 ) if α ∈ (1, n), 4π n-1 2 Γ( n-1 2 ) Γ (k+ n-1 2 ) Γ(k+ n-1 2 ) - Γ ( n-1 2 ) Γ( n-1
2 )

if α = 1. for some critical mass m * > 0 (depending on n, α, β and A).

Case 2: α = β. For any k, f k is either decreasing or increasing or constant (depending on the size of A). If none of them is decreasing, then for any k, f k ≥ f k (0) = 1, so S * = S * = (0, +∞).

Otherwise the same arguments as in case one shows again that S * = (0, m * ) and S * = (0, m * ],

for some critical mass m * > 0 (depending on n, α, β and A).

Cases 3, 4 and 5: α < β. A simple study of the sign of f k shows that each f k is decreasing from 0 to a point γ k , then increasing from γ k to +∞, and we have:

γ k = 1 + α Aβ(1 + β) (µ α k -µ α 1 ) 1 β-α . (1.A.2)
Another simple computation shows that

min f k = f k (γ k ) = 1 - 1 + α Aβ(1 + β) 1+α β-α β -α 1 + β (µ α k -µ α 1 ) 1+β β-α λ k -λ 1 .
(1.A.3)

We must treat the subcases α > 1, α = 1 and α < 1 separately.

Subcase one: α > 1. We use the following classical Stirling formula: We conclude as above.

Subcase three: α < 1. Once again we use the asymptotics As the sequence (µ α k ) is increasing we get )).

Γ(k + n-α 2 ) Γ(k + n-2+α 2 ) ∼ k→∞ k 1-α , to nd that (µ α k -µ α 1 ) 1+β β-α λ k -λ 1 ∼ k→∞ (k 1-α ) 1+β β-α k 2 = k
f k+1 (γ k ) -1 ≤ -γ 1+α k µ α k -µ α 1 λ k+1 -λ 1 + γ 1+β k Aβ λ k+1 -λ 1 = (f k (γ k ) -1) λ k -λ 1 λ k+1 -λ 1 -→ k→∞ -∞.
We are left to show that lim inf T →∞

V α,T (u T ) ≥ V α (u).

(2.2.9)

Setting δ T = (2π) n C(n,α) (V α,T (u T ) -V α (u)), (2.2.9) is equivalent to lim inf Proof of the lim sup inequality for Γ-convergence. Let u ∈ X . We need to nd a sequence (u T ) T >0 of X that converges to u weakly in H 1 (R n ), such that lim sup

T →∞ F ε,T (u T ) ≤ F ε (u).
(2.2.17)

We simply set u T := u, for every T > 0. We only need to verify that lim sup

T →∞
V α,T (u T ) ≤ V α (u).

We procede to the same computations as for the liminf inequality, using S 1 , S 2 , S 3 , I 1 and I 2 as above. We deal with the terms S 3 , I 1 and I 2 exactly as above. The term S 1 is 0 here. For S 2 , we use the isometry property of the Fourier transform to write

S 2 (T, R) = 2π T n | 2kπ T |>R 2kπ T -α ûT ( 2kπ T ) 2 ≤ 2π T n | 2kπ T |>R R -α ûT ( 2kπ T ) 2 ≤ (2π) n R α (-T 2 , T 2 ) n u 2 T = (2π) n R α R n u 2 for T large enough.
This is enough, with the previous estimates, to conclude that (2.2.17) holds. This concludes the proof.

Step three. As the nal step, we discretize the variational problem. In this step, we will use the functional space X := u ∈ H 1 ((-T /2, T /2) n ) :

(-T 2 , T 2 ) n u = m ,
the inmum of the energy is attained by a nite number of balls (of possibly dierent volumes)

innitely fare away from each other, see [START_REF] Bonacini | Local and global minimality results for a nonlocal isoperimetric problem on R N[END_REF]Theorem 2.12]. Our numerical results suggest that in dimension 2, the conjecture holds for any α ∈ (0, 2) (i.e. the whole admissible range). Note that if the conjecture holds, we can compute explicitely the mass m 1 (n, α) > 0 such that there is a minimizer in (1. ).

Note that kf ( m k ) is the energy of k balls of volume m/k, innitely far away from each other. Using the homogeneity of P and V α we nd that m k = |B| (k + 1)

1 n -k 1 n (k) -α n -(k + 1) -α n P (B) V α (B) n 1+α . (2.3.1)
We also set m 0 = 0. The sequence (m k ) is increasing. Then, an equivalent formulation of conjecture 2.3.1, generalized to any dimension n and parameter α ∈ (0, n), is: In particular, as long as there is a minimizer in (1.1.2), it is a ball. When there is no minimizer, the inmum of the energy is attained by a nite number of balls of the same volume, innitely far away from each other.

To get minimizers of (1.1.2) for dierent volume constraint, we set the volume constraint to 1 and add a constant c m to the term V α . Indeed, minimizing inf E⊂R n ,|E|=1 (2.3.2)

P (E) + c m V α (E)
The choice of T is made so that, if 1 N B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF] is the discretization of the ball of volume 1 with side step T N , we have

V α,T (1 N B[1] ) -V α (B[1]) V α (B[1]) ≤ 1%.
Meanwhile, given the number of discretization points N = 2 11 , we cannot increase T too much, otherwise the discretization of candidate minimizers is less and less precise.

For instance, for α = 1 and n = 2, we have Here we explain how we compute V α (B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]) numerically, as needed in Section 2.3 to choose the value of T . In order to compute numerically the improper integral

V α (B[1]) = B[1]×B[1]
dxdy |x -y| 2-α , we add a small term ε > 0 to the denominator of the integrand. So we compute

V α,ε (B[1]) = B[1]×B[1]
dxdy |x -y| 2-α + ε .

To control the error introduced by the parameter ε, we need to estimate the dierence ∆ ε := V α (B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]) -V α,ε (B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]). We have

∆ ε = B[1]×B[1]
dxdy |x -y| 2-α -B[1]×B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF] dxdy |x -y| 2-α + ε = B[1]×B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF] εdxdy |x -y| 2-α (|x -y| 2-α + ε) . ), for some r > 0. This last bound attains its minimum for r = (2-α)ε απ α 2 1 2

. From there we deduce

∆ ε ≤ 2π α 2 2 -α 2 -α α α 2 ε π α 2 α 2
.

With α = 1, we get ∆ ε ≤ 4π Now the proper integral V α,ε (B [START_REF] Adelstein | Morse theory for the uniform energy[END_REF]) can be expressed in polar coordinates, and computed with arbitrary precision in the Julia language, using the HCubature package.

  Choksi et Peletier en dimension 2 : dans ce cas les minimiseurs du modèle de Gamow semble être des boules si ils existent. Introduction a Obstacle problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b Elastic-plastic torsion problem and medial axis . . . . . . . . . . . . . . . . . . c Distance function and cut locus . . . . . . . . . . . . . . . . . . . . . . . . . . . d A variation of the elastic-plastic torsion problem on manifolds . . . . . . . . . . e A non-local isoperimetric problem . . . . . . . . . . . . . . . . . . . . . . . . . .

  e v d m = v m . Theorem b.1. Let us dene F(v) := Ω |∇v| 2 -mv, K 1 := {v ∈ H 1 0 (Ω) : |∇v| ≤ 1}, K 2 := {v ∈ H 1 0 (Ω) : v ≤ d ∂Ω }.

c

  Distance function and cut locus Let M be a smooth Riemannian manifold without boundary of dimension n ≥ 2, b a point of M , d the distance function of M and d b := d(•, b) the distance function to the point b. We will now introduce the notion of cut locus, which is closely related to the function d b . Let us rst recall some notions of Riemannian geometry. Denition c.1. Let γ : [a, b] → M be a geodesic, and J : [a, b] → T M a vector eld along γ.

  c. Distance function and cut locus Denition c.3. The cut locus of b in M , denoted by Cut b (M ), is dened as the set of all cut points of b.

Remark c. 7 .

 7 This denition is compatible with the previous denition of semiconcavity for functions on R n . In [60, Proposition 3.4], C.Mantegazza and A.Mennucci showed that the distance function is a viscosity solution of the Hamilton-Jacobi equation |∇d b | = 1.

  2.10), C.Mantegazza, G.Mascellani and G.Uraltsev used this fact along with geometric measure theory tools to give a complete description of the distributional Hessian and Laplacian of d b . They made use of the fact that the gradient of semiconcave functions has bounded variations, in particular. They proved the following.

D 2 d

 2 b is the classical Hessian of d b wherever d b is smooth (i.e. almost everywhere), Vol is the Riemannian volume measure, ν is a unit normal vector to Cut b (M ) wherever Cut b (M ) is smooth (i.e. H n-1 -almost everywhere), ∇d + b and ∇d - b are limits of ∇d b to either side of Cut b (M ), and H n-1 Cut b (M ) is the n -1-dimensional Hausdor measure restricted to Cut b (M ). The distributional Laplacian of d b is given by: ∆d b = ∆d b Vol -∇d + b -∇d - b H n-1 Cut b (M ) , where ∆d b is the classical Laplacian of d b , wherever d b is smooth. Our main result in Chapter 2 is a complement to the description of the non-smooth behavior of the distance function on the cut locus. It deals with the Laplacian of d b in the sense of barriers.

Theorem c. 10 (

 10 Theorem 2.1.11). Let M be a smooth manifold without boundary of dimension n ≥ 2, and b ∈ M . Let d b be the distance function to the point b, and Cut b (M ) the cut locus with respect to b. The Laplacian of d b is -∞ at every point of Cut b (M ), in the sense of barriers.

Our proof relies on the maximum principle, applied to a function involving ∇u d m 2 ,

 2 and the Euler-Lagrange equation ∆u d m = -2m. For the elastic-plastic torsion problem, as mentioned earlier, we know that the elastic set contains the medial axis, and the former converges to the latter in the Hausdor sense. For problem (d.1), we have the analogous result: Theorem d.4 (Theorem 3.1.2). We have Cut b (M ) ⊂ {|∇u m | < 1}, and the following convergence holds in the Hausdor sense:

function d b

  is locally semiconcave on M \ {b}. This implies that, as convex functions, d b has a well dened generalized gradient at every point see Appendix A.1, Proposition A.1.6. At a point p where d b is not dierentiable, the norm of its gradient is given by |∇d b | (p) := max(0, sup v∈TxM,|v|=1 ∂ + v d b (p)). As shown in Appendix A.1, Proposition A.1.10, if there exist two minimizing geodesics from p to b, then |∇d b (p)| < 1. In particular, we have Cut b (M ) = {|∇d b | < 1}.

u 2 m

 2 can be used as an approximation of Cut b (S) λ . In turn, we can see that Cut b (S) λ is a good approximation of Cut b (S). First, we have the following convergence in the Hausdor sense: Cut b (S) λ -→ λ→0 Cut b (S).

d

  b and Cut b (M ), we prove a proposition about the behavior of d b on Cut b (M ), that is later used in Chapter 3. More specically, we prove: Theorem (Theorem 2.1.11). Let M be a smooth manifold without boundary of dimension n ≥ 2, and b ∈ M . Let d b be the distance function to the point b, and Cut b (M ) the cut locus with respect to b. The Laplacian of d b is -∞ at every point of Cut b (M ), in the sense of barriers. This means that for any p ∈ Cut b (M ), for any A > 0, there exists a smooth function φ dened in a neighborhood of p such that φ ≥ d b , φ(p) = d b (p) and ∆φ(p) ≤ -A.

  The main tool isKorevaar's convexity maximum principle [54, Theorem 1.3]. Let us present this result. Let f : Ω → R be a C 2 function satisfying an equation of the form P (x, f (x), ∇f (x), Hf (x)) = 0 in Ω, for a certain elliptic operator P . The convexity function of f is the function c dened for any [x, y] ⊂ Ω and λ ∈ [0, 1] by c(x, y, λ)

  stating and proving the result of this chapter, let us introduce and give some context about the notions of distance function and cut locus. In this chapter, M is a smooth Riemannian manifold without boundary of dimension n ≥ 2, b a point of M , d the distance function of M and d b := d(•, b) the distance function to the point b. We are interested in the properties of the function d b . Before giving an account of these properties, let us recall some notions of Riemannian geometry. Denition 2.1.1. Let γ : [a, b] → M be a geodesic, and J : [a, b] → T M a vector eld along γ.

Denition 2 . 1 . 3 .

 213 The cut locus of b in M , denoted by Cut b (M ), is dened as the set of all cut points of b.

[ 74 ,Proposition 2 . 1 . 6 .

 74216 Chapter III, Section 4, proposition 4.5] for a precise statement. The function d b is 1-Lipschitz. At every point p where d b is dierentiable, ∇d b is given by the speed vector of the minimizing unit speed geodesic from b to p. In particular, we have |∇d b | = 1 at every point where d b is dierentiable.

Remark 2 . 1 . 8 .

 218 For a function u dened on an open set of R n , u is C-semiconcave if and only if the map x → C |x| 2 -u(x) is convex. In [60, Proposition 3.4], C.Mantegazza and A.Mennucci showed that the distance function is a viscosity solution of the Hamilton-Jacobi equation |∇d b | = 1.

  D 2 d b is the classical Hessian of d b where d b is smooth (i.e. almost everywhere), Vol is the Riemannian volume measure, ν is a unit normal vector to Cut b (M ) where Cut b (M ) is smooth (i.e. H n-1 -almost everywhere), ∇d + b and ∇d - b are limits of ∇d b to either side of Cut b (M ), and H n-1 Cut b (M ) is the n -1-dimensional Hausdor measure restricted to Cut b (M ). The distributional Laplacian of d b is given by:

  ∆d b is the classical Laplacian of d b where d b is smooth. The main result of this Chapter is a complement to the description of the non-smooth behavior of the distance function on the cut locus. We are interested in the Laplacian of d b in the sense of barriers. Denition 2.1.10. Let ψ : M → R be a continuous function, and p ∈ M . We say that the Laplacian of ψ at p is -∞ in the sense of barriers if for any A > 0, there exists a smooth function φ : M → R dened on a neighborhood of p such that φ ≥ ψ, φ(p) = ψ(p) and ∆φ(p) ≤ -A. We prove the following. Theorem 2.1.11. Let M be a smooth manifold without boundary of dimension n ≥ 2, and b ∈ M . Let d b be the distance function to the point b, and Cut b (M ) the cut locus with respect to b. The Laplacian of d b is -∞ at every point of Cut b (M ), in the sense of barriers. This result proves useful when one wants to apply the maximum principle to a function involving the distance function d b . It will be applied to the study of a variational problem in Chapter 3.

2 . 2

 22 Proof of Theorem 2.1.11

Figure 2 . 1 :

 21 Figure 2.1: Construction of r, R, q, p ε , r ε , f , g, g ε .

2

 2 is an integer, M a smooth n-dimensional compact Riemannian manifold without boundary, b ∈ M any point of M (that can be thought of as a base point), and d b the distance function to the point b in M . Denition 3.1.1. The cut locus of b in M can be dened as the closure of the set of points p ∈ M such that there exists at least two minimizing geodesics between p and b. We will denote it by Cut b (M ). Equivalently, it is also the set of points around which the distance function to the point b -denoted by d b -is not smooth.We refer to the introduction of Chapter 2 for a more detailed introduction to the notion of cut locus.The goal of this Chapter is to study an extension of the elastic plastic torsion problem to manifolds, and its relation with the cut locus of b in M . We are interested in the following variational problem.

Figure 3 . 1 :

 31 Figure 3.1: A circle with its medial axis, and a polygonal approximation of a circle, with its medial axis.

( 1 . 1 . 1 )

 111 can be replaced with an obstacle-type constraint where the obstacle is the distance function to the boundary of Ω, i.e. the function d ∂Ω dened by d ∂Ω (x) := min |x -y| : y ∈ ∂Ω .

60 ,

 60 Proposition 3.4] that the function d b is locally semiconcave on M \ {b}. This implies that, as convex functions, d b has a well dened generalized gradient at every point see Appendix A.1, Proposition A.1.6. At a point p where d b is not dierentiable, the norm of its gradient is given by |∇d b | (p) := max(0, sup v∈TxM,|v|=1 ∂ + v d b (p)).

( 3 . 1 . 4 )

 314 As shown in AppendixA.1, Proposition A.1.10, if there exist two minimizing geodesics from p to b, then |∇d b (p)| < 1. In particular, we have Cut b (M ) = {|∇d b | < 1}.

m→+∞

  Cut b (M ) in the Hausdor sense.

  d(x, Cut b (M ) λ ) -→ m→+∞ 0, and sup x∈Cut b (M ) λ+ε d(x, E m,λ ) -→ m→+∞ 0.

( 3 .

 3 1.11) In addition to its own theoretical interest, this theorem is used in Chapter 4 to compute numerically a stable approximation of the cut locus of a closed surface.

Remark 3 . 1 . 3 .

 313 We could have replaced the point b with a smooth hypersurface S ⊂ M in the above problem. However, our interest here is in the instability of the cut locus with respect to the metric of M , and this aspect is already present with a point b instead of S. So for simplicity, we focus on the case of the cut locus of a point.The rest of the Chapter is organized as follows. In Section 3.2, we prove the C 1,1 regularity of u m . In Section 3.3, we prove that the analogous identity to (3.1.2) does not hold in general for problem (3.1.1), but it holds for m large enough. In Section 3.4, we prove the assertions (3.1.8) and (3.1.9) of Theorem 3.1.2. In Section 3.5, we prove the assertions (3.1.10) and (3.1.11) of Theorem 3.1.2.

( 3 . 2 . 1 )

 321 It is the same problem as(3.1.1), where the constraint on the gradient has been replaced with an obstacle-type constraint where the obstacle is the distance function d b . In this section, we prove the C 1,1 regularity of u d m . In the next section, we will prove that for m large enough we have u d m = u m . Thanks to Theorem 2.1.11, we can prove the following.Lemma 3.2.1. For any m > 0, there exists a functiond b that is smooth on M \ {b}, such that u d m ≤ d b ≤ d b on M and d b < d b on Cut b (M ). In particular, u d m is also the solution of inf M |∇u| 2 -mu : u ∈ H 1 (M ), u ≤ d b . (3.2.2)Proof. It is a consequence of the minimality of u m in (3.2.1) that we have in the sense of distributions:

  Thus, the function u m -d b is upper semi-continuous and negative on the compact set Cut b (M ), so there exists ε > 0 such that the set {u m < d b -ε} is an open set containing Cut b (M ). Now, choose any smooth bump function ρ : M → [0, 1] that is compactly supported in {u m < d b -ε} \ {b} and such that ρ = 1 on Cut b (M ), and a smooth function d b,ε : M → R such that d b -ε ≤ d b,ε < d b .

  b := (1 -ρ)d b + ρd b,ε .

( 3 . 2 . 4 )

 324 This function is smooth on M \ b because d b is smooth on the complement of Cut b (M ) ∪ {b}. The condition d b ≤ d b is a consequence of d b,ε < d b . The inequality d b < d b on Cut b (M ) is also a consequence of d b,ε < d b , plus the fact that ρ = 1 on Cut b (M ). We are left to show that u m ≤ d b . On the complement of the support of ρ, we have d b = d b , so u m ≤ d b . On the support of ρ, (3.2.4) and (3.2.3) imply

Lemma 3 . 2 . 2 .

 322 For any m > 0, the function u d m is continuous on M . Proof. We will reduce our problem to a classical obstacle type variational problem on an open subset of R n , by a series of elementary modications, and apply a classical W 2,p regularity theorem. From lemma 3.2.1, we know that there exists an open set U ⊂ M and ε > 0 such that Cut b (M ) ⊂ U and u d m ≤ d b -ε on U . On U , u d m veries the Euler-Lagrange equation of (3.2.1), i.e ∆u d m = -2m. In particular, it is smooth on U . Let Ω ⊂ M be a smooth open set such that U c ⊂ Ω, ∂Ω ⊂ U and Cut b (M ) ∩ Ω = ∅.As U c ⊂ Ω, it suces to show that u d m is continuous on Ω. As ∂Ω ⊂ U , u d m is smooth on ∂Ω, so there exists a smooth function v m on Ω such that v m = u d m on ∂Ω. Then, one can check that u d m is a solution of the following variational problem:

  can now dene the set E d m := {u d m < d b }, for any m > 0. It is an open subset of M , on which u d m veries the Euler-Lagrange equation of problem (3.1.1): ∆u d m = -2m. We can now prove the following lemma. Lemma 3.2.3. For any m > 0, we have u d m = d b in a neighborhood of b. Proof. Let us assume that we have constructed a C 1 function v on B(b, R) for some R > 0, such that

  and so x ∈ B(b, R). Hence the function v -u d m attains a positive maximum inside the open set E d m ∩ B(b, R), but its Laplacian veries in the distributional sense: ∆(v -u d m ) = ∆v + m ≥ 0, (3.2.11) which yields a contradiction by the maximum principle. Then, the maximum of v -u d m on B(b, R) is non-positive, and we get u d m ≥ v = d b in B(b, ε), which concludes the proof.

( 3 . 2 . 15 )

 3215 Note that by applying the Laplacian formula in polar coordinates to the distance functiond b (x) = r, we nd that ∆d b = ∂ r det g 2 det g .

( 3 . 2 . 16 )

 3216 Because of lemma 3.2.5, we also have∆d b (x) = n -1 r + o(1).With (3.2.15) and (3.2.16), this last equation yields in the classical sense ∆v = -m 2 + o(1)f (r) on B(b, R) \ B(b, ε).

Figure 3 . 2 :

 32 Figure 3.2: Curve γ generating M by rotating around the z-axis.

3 . 7 ) 8 )

 378 by the construction of r. Now we use the fact u d m = u m , which means that ∇u d m ≤ 1, and so |ρ m | ≤ 1. With the running assumption ρ m (4) ≥ 3.5, this implies ρ m (2) ≥ 1.5. As ρ m (1) ≤ 1, we get ρ m (2) -ρ m (1) ≥ 0.5. Then, (3.3.7) and (3.3.6) yield F(ρ) -F(ρ m ) ≤ 16 • 10 -10 -0.25 Recalling that we have chosen m = 10 -10 , it contradicts the minimality of ρ m in (3.3.4).

3 . 5 )

 35 . Thus (3.3.6) becomes instead: F(ρ) -F(ρ m ) ≤ 16 • 10 -10 -Recalling that we have chosen m = 10 -10 , T = 10 10 and r ≥ 1 between 5 and T -1, it contradicts the minimality of ρ m in (3.3.4). This concludes the proof. Proof of Theorem 3.3.2. Recall that the function u d m is C 1 except at b, by Proposition 3.2.4. Let x = b be in the contact set P d m := {u d m = d b }. By the lemma 3.2.1, we have x / ∈ Cut b (M ), and so the distance function d b is dierentiable at x. It is a simple consequence of the constraint u d m ≤ d b and the equality u d m (x) = d b (x) that we have ∇u d m (x) = ∇d b (x). The desired inequality ∇u d m (x) ≤ 1 follows. In the non-contact set E d m = {u d m < d b }, the function u d m veries the Euler-Lagrange equation ∆u d m = -2m.

( 3 . 3 . 9 ) 2 = 2 + 2

 339222 In particular it is smooth, and we may apply the Bochner-Weitzenböck formula:∆ ∇u d m 2Ric(∇u d m , ∇u d m ) + 2 D 2 u d m (∇∆u d m , ∇u d m ),(3.3.10) 3.4 Convergence of the non-contact set In this section we show that the non-contact set E d m = {u d m < d b } Hausdor-converges to Cut b (M ).

Lemma 3 . 4 . 1 . 1 .

 3411 We have d b -u d m L ∞ (M ) ≤ C/m, for some positive constant C depending on M only. Proof. We only need to prove the proposition for m large enough. Therefore, thanks to Theorem 3.3.2, we will assume that m is large enough so that ∇u d m ≤ We only need to show the estimate on E d m since outside this set, u d m and d b are the same. We will show that for m large enough, we have ∀p ∈ E d m , ∃p ∈ (E d m ) c such that d(p, p) < 5n/m.

( 3 . 4 . 1 ) 2 .

 3412 This will conclude the proof since by the 1-Lipschitzianity of u d m and d b , we then haved b (p) -u d m (p) ≤ d b (p) -u d m (p) + 2d(p, p) = 0 + 2d(p, p) ≤ 5n m ,which is what we need. In order to prove (3.4.1), we argue by contradiction and assume that B 5n/m (p) ⊂ E d m . We want to apply the maximum principle to the function v dened on B 5n/m (p) by the following formula v(p) := u d m (p) -inf For any p ∈ B 5n/m (p), we have ∆u d m (p) = -2m because we have assumed B 5n/m (p) ⊂ E d m .

Therefore, we obtainm

  on B 5n/m (p) ⊂ E d m , So we can apply the maximum principle to v to get v(p) ≥ inf ∂B 5n

( 3 . 4 . 2 ) 3 . 4 . 1 , 2 ,

 3423412 Convergence of the non-contact set As we have taken m large enough so that ∇u d m ≤ which contradicts the estimate (3.4.2). This concludes the proof. Theorem 3.4.2. For any m > m > 0, we have Cut b (M ) ⊂ E d m ⊂ E d m , and E d m -→ m→∞ Cut b (M ) in the Hausdor sense.

D

  [ac] D b D d u m ≥ -C M D 2 u m . (3.5.14) By symmetry of the tensor D 2 u m , we have D c D a D [bd] u m = 0.

  5.12), (3.5.13), (3.5.14) and (3.5.15) together, we nd

( 3 . 5 . 16 )

 3516 With this estimate, the estimate (3.5.11) yields

( 3 . 5 . 20 )

 3520 where we have used that u m is bounded by the diameter of M since u m ≤ d b . So we may transform(3.5.19) into

  to conclude that ∀x ∈ ∂E m , lim y→x y∈Em D 2 u m (y) ≤ C d Id.

( 3 . 5 . 22 )

 3522 The minimality of u m in (3.2.1) implies-∆u m -2m ≥ 0, u m ≤ d b and (-∆u m -2m)(u m -d b ) = 0.(3.5.23)

( 3 . 5 . 29 )

 3529 Inequalities (3.5.27), (3.5.28) and (3.5.29) yield a contradiction. So (3.5.26) is true.

( 3 . 5 . 4 .

 354 26) translates into µ ≤ C d + 1 (3.5.30) Furthermore, inside E m we have ∆u m = -2m, so λ + µ = -2m.

( 3 . 5 . 31 )

 3531 For m large enough, we see that (3.5.30) and (3.5.31) imply

u m 2 .

 2 The second term is null because it contains D b (Y c Y d ) = (D b Y c )Y d + Y c D b Y d , and DY = 0. The third term is also null. By our preceding remark (3.5.34) about D 2 Y , the fourth term is bounded from below by -C M -D 2 Finally, the rst term is treated just like in the preceding proof. Recall the estimate (3.5.16):

Proposition 3 . 5 . 6 .

 356 Let λ > 0. Let Cut b (M ) λ and E m,λ be dened as in (3.1.6) and (3.1.7). We have sup p∈E m,λ d(p, Cut b (M ) λ ) -→ m→+∞ 0, (3.5.36) and sup p∈Cut b (M ) λ+ε d(p, E m,λ ) -→ m→+∞ 0. (3.5.37) Proof. Let us assume by contradiction that (3.5.36) does not hold. Let (m k ) k≥0 and (p k ) k≥0 be sequences in (0, ∞) and M respectively, such that m k -→ k→+∞ +∞ and for some η > 0, for any k ≥ 0, p k ∈ E m k ,λ and d(p k , Cut b (M ) λ ) > η.

( 3 . 5 . 38 )

 3538 Up to taking subsequences, we may assume that (p k ) converges to a point p ∞ in M . As the functions (u m ) m>0 are uniformly semiconcave and converge to d b as m goes to ∞ (Proposition 3.5.5 and lemma 3.4.1), we get from Proposition A.1.8:

Proposition 4 . 2 . 1 .

 421 The map λ → Cut b (S) λ is non-increasing, and Cut b (S) = λ>0 Cut b (S) λ .

The geodesics γ 1

 1 and γ 2 verify the geodesic equation: γl + Γ l ij γi γj = 0, where the Γ l ij are the Christoel symbols of the metric of S. In particular, by continuity of the solution of a second order dierential equation with respect to initial conditions, if the angle between γ 1 and γ 2 at their common starting point is small, then their spread is also small. Therefore, applying [28, Proposition 3.4], we deduce that if λ has been taken small enough, then any point x ∈ Cut b (S) \ Cut b (S) λ is a tree point of Cut b (S). It remains to show that Cut b (S) λ is closed to conclude that it is a deformation retract of Cut b (S) and conclude the proof. But this is a consequence of the fact that d b is semiconcave, and the upper semicontinuity of the generalized gradient of convex functions (see also Proposition A.1.8 for instance).

Theorem 4 . 3 . 1 (

 431 Theorem 3.1.2). There exists m 0 > 0 such that for any m > m 0 , the function u m is locally C 1,1 on S \ {b}. For any m > m > m 0 ,Cut b (S) ⊂ {|∇ S u m | < 1} ⊂ {|∇ S u m | < 1}.

  S u m | < 1} -→ m→+∞ Cut b (S) in the Hausdor sense.

( 4 . 3 . 2 )

 432 Finally, for any ε > 0, sup x∈E m,λ d(x, Cut b (S) λ ) -→ m→+∞ 0, and sup x∈Cut b (S) λ+ε d(x, E m,λ ) -→ m→+∞ 0.

( 4 . 3 . 3 )

 433 Therefore, we can use E m,λ as an approximation of Cut b (S) λ . In conclusion, we will use E m,λ as an approximation of Cut b (S).

  .11)], and the fact that, following the notations of [31, lemma 4.1], we have ν 2 n+1 = 1 -j≤n ν 2 j . The estimate on D 2 a follows from the same equations, plus the identity D 2 a(x) = -2∇d(x)D 2 d(x). Dening F (u) = S |∇ S u| 2 -mu , we have Proposition 4.4.2. Let u m,h be the solution of problem (4.4.6) for k

( 4 .

 4 4.11) step 2. Symmetrically, let u m,h the solution of the discrete problem (4.4.6), u l h

( 4 .

 4 4.12)

( 4 . 4 . 13 )

 4413 As in the proof of Proposition 4.4.2, using the relation u

Figure 4 . 1 :

 41 Figure 4.1: Three dierent views of the approximation of a cut locus on a standard torus

Figure 4 . 2 :

 42 Figure 4.2: Three dierent views of the approximation of a cut locus on a standard torus, without representing the surface

Figure 4 . 4 :

 44 Figure 4.4: Three dierent views of the approximation of a cut locus on a torus of genus 2, without representing the surface

Figure 4 .

 4 5 and 4.6 represent the voronoi diagrams obtained with 10, 30 and 100 points for surfaces of genus 2 and 3. The expected computational complexity is exactly of the same order as with a single source point.

Figure 4 . 5 :

 45 Figure 4.5: Approximation of the voronoi cells on a torus of genus 2 of 10, 30 and 100 points. Every column represent two dierent views

Figure 4 . 6 : 1 2

 461 Figure 4.6: Approximation of the voronoi cells on a torus of genus 3 of 10, 30 and 100 points. Every column represent two dierent views

a 1 .•

 1 ..a k b 1 ...b l 2 = T a 1 ...a k b 1 ...b l T b 1 ...b l a 1 ...a k . • Laplacian: ∆v = g ab D a D b v. Product rule: D a (T a 1 ...a k b 1 ...b l S c 1 ...c i b 1 ...b j ) = (D a T a 1 ...a k b 1 ...b l )S c 1 ...c i b 1 ...b j +T a 1 ...a k b 1 ...b l (D a Sc 1 ...c i b 1 ...b j

D

  a (g bc T a 1 ...a k b 1 ...b l ) = g bc D a T a 1 ...a k b 1 ...b l . • Riemann curvature tensor: D 2 [ab] Z c = R c abd Z d . Part II A non-local isoperimetric problem 97 Outline of part II In this part, we study a non-local isoperimetric problem derived from Gamow's liquid drop model for the nucleus. More precisely, we are interested in min E⊂R n ,|E|=m

1 n

 1 (1.1.3), such that |E m | = m, and let E * m be the rescaling of E m of the same mass as the unit ball B centered at 0 (ie E * m = |B| m E m ). Then, up to modifying the sets (E m ) by some sets of measure 0, the boundaries of the sets (E * m ) Hausdor-converge to the boundary of B as m → +∞.

103 1 . 2 .

 12 Existence of a minimizer in(1.1.3)

( 1 . 4 . 1 )

 141 where we have set γ := m |B| 1 n and E * := 1 γ E. As the unique volume constrained minimizer of U β,A is the ball B[m], this implies that if E is a minimizer of E α,β,A at mass m for m large, U β,A (E) must be close to U β,A (B[m]). This in turn will imply that E is close to B[m]. Note that according to the rescaling (1.4.1), proving Theorem 1.1.1 is equivalent to proving that if (E γ ) γ>0 is a family of Borel sets such that |E γ | = |B|, and each set E γ is a volume-constrained minimizer of the functional E α,β,A,γ :

  B) -→ γ→∞ 0, which concludes the proof thanks to Lemma 1.4.2. We are now in position to prove Theorem 1.1.1. Proof of Theorem 1.1.1. Let (E γ ) γ>0 be a family of Borel sets such that |E γ | = |B|, and each set E γ is a volume-constrained minimizer of the functional E α,β,A,γ :

  This last estimate with (1.4.8) and (1.4.10) gives

( 1 .

 1 4.15) i.e. for any non-negative measurable function f , S f (r)r n-1 dr = (1/2,u) f (T (r))r n-1 dr.

  have ∀r ∈ (1/2, u), T (r) ≥ r.Proof. The existence and uniqueness of the map T is given by the result[START_REF] Santambrogio | Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling[END_REF] Theorem 2.5].Only the fact that T (r) ≥ r needs to be checked. For r ∈ (1/2, u),taking f = 1 (1/2,T (r)) in (1.4.22) yields S∩(1/2,T (r)) t n-1 dt = (1/2,u) 1 T (t)≤T (r) t n-1 dt.Let us argue by contradiction and assume that T (r) < r. As T is non-decreasing, the previous equality yieldsS∩(1/2,T (r)) t n-1 dt ≥ (1/2,r) t n-1 dt > (1/2,T (r)) t n-1 dt ≥ S∩(1/2,T (r))

( 1 . 4 .

 14 21) becomesU β,A (E u ) -U β,A (E) = ∂B ( 1 2 ,u(x))

( 1 . 4 . 24 )

 1424 To estimate(1.4.23) and(1.4.24), we use the two following inequalities:∀x ∈ ∂B, ∀s ≥ r > 1 2 , r β -s β ≤ -C(β) |r -s| , (Φ α Eu + Φ α E )(rx) -(Φ α Eu + Φ α E )(sx) ≤ C(n, α) |r -s| ,where the second inequality comes from Lemma 1.4.5. With these and (1.4.23) and (1.4.24), we getγ 1+α V α (E u ) + γ 1+β U β,A (E u ) -γ 1+α V α (E) + γ 1+β U β,A (E) ≤ ∂B (1/2,u(x))γ 1+α C(n, α) -γ 1+β AC(β) |r -T x (r)| r n-1 drdH n-1 (x).

[u] 2 1-α 2 ≤ 2 ( 4 ∂B×∂B

 2224 ∂B×∂B |u(x)| 2 + |u(y)|) 2 |x -y| n-α dH n-1 (x)dH n-1 (y) = |u(x)| 2 |x -y| n-α dH n-1 (x)dH n-1 (y) = 4 ∂B ∂B dH n-1 (y) |x -y| n-α |u(x)| 2 dH n-1 (x) = C(n, α) ∂B |u(x)| 2 dH n-1 (x) = C(n, α) u 2 L 2 (∂B) , so that (1.4.28) gives V α (E t ) ≥ V α (B) -t 2 2 C(n, α) u 2 L 2 (∂B) -C(n, α)t 3 u 2 L 2 (∂B) .

A 1 0A( 1 +A ( 1 ++ β dH n- 1 . 3 ∂B u 3 dH n- 1 ≥

 111131 U β,A (E u ) + γ 1+α C(n, α, β, A) u 2 L 2 (∂B) -γ 1+β C(n, α, β, A) u 2 L 2 (∂B) ,which in turn implies(1.4.26) for γ large enough. The estimate (1.4.15) is now a consequence of (1.4.25) and (1.4.26). The theorem results from (1.4.15) and the isoperimetric inequality.Proof of Lemma 1.4.7. The proof of (1.4.28) is given in[33, equation (5.20)], under the hy-pothesis u C 1 (∂B) ≤ 1 instead of u L ∞ (∂B) ≤ 1.However it is clear from the proof that it holds also for u L ∞ (∂B) ≤ 1 only. (The reason why it was stated with the stronger hypothesis u C 1 (∂B) ≤ 1 is because it is needed to get the corresponding estimate for the perimeter.)Let us prove(1.4.27). Using spherical coordinates, we can computeU β,A (E t ) = ∂B 1+tu(x) 0 |rx| β r n-1 drdH n-1 (x) = ∂B tu(x)) n+β r n+β-1 drdH n-1 (x) = ∂B tu) n+β n Setting h(t) := ∂B (1 + tu) n+β dH n-1 , we then have U β,A (E t ) -U β,A (B) = A n+β (h(t) -h(0)).The Taylor-Lagrange inequality gives (1 + tu) n+β ≥ 1 + (n + β)tu + (n + β)(n + β -∂B tudH n-1 + (n + β -1) ∂B (tu) 2 2 dH n-1 -C(n, β)t 3 u 2 L 2 (∂B) .

( 1 . 4 . 29 )∂B( 1 + 2 2 L 2 -µ α 1 λ k -λ 1 + γ 1+β Aβ λ k -λ 1 .

 14291222111 Now we use the volume constraint |E t | = |B| to estimate tu. The volume constraint can be expressed as tu) n dH n-1 = dH n-1 -C(n)t 3 u (∂B) (1.4.30) This with (1.4.29) gives (1.4.27). 1.A Appendix: study of the sets S * and S * from the proof of Proposition 1.3.3 In this appendix, we give explicit forms of the sets S * and S * , needed in the proof of Proposition 1.3.3. Let us dene, for all k ≥ 2, a function f k : (0, ∞) → R by f k (γ) = 1 -γ 1+α µ α k Then, the sets S * and S * from the proof of Proposition 1.3.3 are dened by S * = {∀k ≥ 2, f k > 0} and S * = {∀k ≥ 2, f k ≥ 0}.

( 1 .A. 1 )

 11 Recall also that for any k ≥ 0, λ k = k(n + k -2). Now let us treat each case enumerated in Proposition 1.3.3 separately.Case 1: α > β. A simple study of the sign of f k shows that each f k is increasing from 0 to a point γ k , then decreasing from γ k to +∞. Furthermore, f k (0) = 1 and lim +∞ f k = -∞, so each f k has exactly one zero and is positive left of this zero and negative right of it. At last, for any constant K > 0, f k (γ) -→ k→∞ 1 uniformly in γ ≤ K. Putting these facts together shows that the sets S * and S * have the forms: S * = (0, m * ) and S * = (0, m * ], 119 1.A. Appendix: study of the sets S * and S * from the proof of Proposition 1.3.3

1 .

 1 α . With (1.A.1), this means that the sequence (µ α k ) is bounded. As λ k -→ k→∞ ∞, we get from (1.A.3) min f k -→ k→∞ Thus there exists an index k 0 such that S * = k 0 k=2 {f k > 0} and S * = k 0 k=2 {f k ≥ 0}.

( 1 .A. 4 )

 14 As for any k, lim +∞ f k = +∞, we get that S * and S * both contain an unbounded interval, which is what we wanted. Subcase two: α = 1. We use the classical asymptotics of the digamma function Γ Γ : Γ Γ (x) ∼ x→+∞ ln(x), to nd that, according to (1.A.3), min f k -→ k→+∞ 1.

1 , 1 ,α k+1 -µ α 1 λ

 111 once again we have min f k -→ k→+∞ and we conclude as above. If β < 1, then we havemin f k -→ k→+∞ -∞ (1.A.6)Also, by denition we havef k+1 (γ k ) -1 = -γ 1+α k µ k+1 -λ 1 + γ 1+β k Aβ λ k+1 -λ 1 .

( 1 .

 1 A.7)With(1.A.6), this means that[γ k , γ k+1 ] ⊂ ( S * ) c ∩ (S * ) c .Moreover, from (1.A.2), we have γ k -→ k→+∞ +∞, so S * and S * are both bounded, which is what we wanted. At last, if β = 1 we nd using (1.A.5) that there exists a constant C α such that min f k -→ > C α , we conclude as in case one. For A 1+α β-α < C α , we conclude as above that S * and S * are both bounded. For A 1+α β-α < C α , we have to use the following more precise form of Stirling's approximation:

2 -I 2 ≤

 22 k,T (u T )| 2 -|ξ| -α |û(ξ)| |ξ|≤R |ξ| -α |û(ξ)| 2 dξ -|ξ|>R |ξ| -α |û(ξ)| 2 dξ =: S 1 (T, R) + S 2 (T, R) + S 3 (T, R) -I 1 (R) -I 2 (R),with obvious notations. We haveS 2 ≥ 0, R -α |û(ξ)| 2 dξ = (2π) n R -α u 2 ≤ (2π) n R -α u = (2π) n R -α m, (2.2.11)where we used the isometry property of the Fourier transform and the fact that u takes its values in [0, 1]. From (2.2.10) and (2.2.11) we deduce lim infT →∞ δ T ≥ lim inf T →∞ S 1 (T, R) + lim inf T →∞ (S 3 (T, R) -I 1 (R)) -(2π) n R -α m.

( 2 . 2 . 12 )

 2212 As the Fourier transform is 1-Lipschitz from L 1 (R n ) to L ∞ (R n ), and u T → u in L 1 (R n ), we have ∀ξ ∈ R n , | ûT (ξ)| 2 -|û(ξ)| 2 ≤ ε(T ),with ε(T ) -→

  to conclude that S 1 (T, R) -→ T →∞ 0. But this comes from the fact that |ξ| -α dξ.

( 2 . 2 . 14 ),

 2214 Indeed using the monotonicity of x → x -α , we have in dimension 1From this we deduce convergence (2.2.14) in dimension 1, and the convergence in any dimension can be shown with the same idea. In conclusion, we have S 1 (T, R) -→ T →∞ 0 and so (2.2.12) becomes lim infT →∞ δ T ≥ lim inf T →∞ (S 3 (T, R) -I 1 (R)) -(2π) n R -α m.

( 2 . 2 . 15 ) 2 +-α dξ u 2 L 1 + 1 ,

 22152211 Given η > 0 small, we have|S 3 (T, R) -I 1 (R)| ≤ |ξ|≤η |ξ| -α |û(ξ)| 2 dξ . (2.2.16)In the right-hand side of this last inequality, the rst term goes to 0 as T → ∞ by classical convergence of Riemann sums for continuous functions on a nice domain. For the second term, we use the estimate û L ∞ ≤ u L 1 , and convergence (2.2.14), to get lim supT →∞ |S 3 (T, R) -I 1 (R)| ≤ |ξ|≤η |ξ| |ξ|≤η |ξ| -α |û(ξ)| 2 dξ u 2 Land so letting η go to 0, lim supT →∞ |S 3 (T, R) -I 1 (R)| = 0.

1 . 2 )

 12 if and only if m < m 1 . Indeed, given m > 0, let us setf (m) = P (B[m]) + V α (B[m]).Then, dene m k as the unique solution of kf (

Conjecture 2 . 3 . 2 .

 232 If m ∈ [0, m 1 ], the ball of volume m is the unique minimizer of (1.1.2). For m ∈ (m 1 , ∞), there is no minimizer in (1.1.2). Moreover, for any k ∈ N * and m ∈ [m k-1 , m k ], we have inf E⊂R n ,|E|=m P (E) + V α (E) = kf ( m k).

  (a) A = 0 (b) A = 1

Figure 2 . 2 :

 22 Figure 2.2: α = 1, c m = 3.0, β = 16, Ω is a disk.

141 2 .

 2 A. Appendix: computation of V α (B[1])

  The cut locus is quite a fundamental object in Riemannian geometry, and it has been studied by many mathematicians from varied backgrounds. See Section 2.1 for more details. It is a dicult object to compute numerically, because it is unstable with respect to C 1 variations of the metric of M . In particular, the cut locus of the polyhedral approximation of a given surface S may be quite larger than the cut locus of S itself.In Chapter 1, we rst study the elastic-plastic torsion problem, in relation with the medial axis. Given a smooth bounded open set of R n , the medial axis of Ω, denoted by M(Ω), is the set of points x ∈ Ω that have at least two closest points on the boundary of Ω, ∂Ω.

  r(x) is the radius of the smallest ball containing the set of all closest points to x on ∂Ω, i.e. the set {z ∈ ∂Ω : |x -z| = d ∂Ω (x)}. Basically, they showed that for λ small enough, M λ (Ω) is a good approximation of M(Ω) and it is stable with respect to variations of the boundary of Ω. Therefore, one can try and compute M λ (Ω) instead of M(Ω), and that is

	.1 in Chapter 1). It makes it non-trivial to compute numerically as well.
	In 2005, Chazal and Lieutier introduced the notion of λ-medial axis to give a theoretical
	framework to tackle the problem of instability. It is dened, for any λ > 0, as
	M λ (Ω) := {x ∈ Ω : r(x) ≥ λ},

where

  For any j ∈ N, the sequence (E k ∩B R j ) k∈N is a sequence of uniformly bounded Borel sets, with uniformly bounded perimeters. Indeed, the inequalities P (E k ∩ B R j ) ≤ P (E k ), P ≤ E α,β,A and (1.2.1) together give

.2.1) Set g(x) = A |x| β . As g(x) -→

|x|→+∞ +∞, we can take a sequence of positive radius (R k ) k∈N and a sequence of positive constants (A j ) j∈N such that A j -→ j→∞ +∞ and for all x / ∈ B R j , g(x) > A j .

  It is a direct consequence of the same theorem for problem (1.1.2) (see[START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] Theorem 1.3]), as balls centered at 0 are also volume-constrained minimizers of U β,A . This last fact is a consequence of Riesz inequality regarding symmetric decreasing rearrangements (see[START_REF] Lieb | of graduate studies in mathematics[END_REF] for rearrangement inequalities). Note that Proposition 1.3.1 is true if we replace the potential A |x| β with a radially symmetric non-decreasing function g. We say that a set E ⊂ R n of volume m is a L 1 -local minimizer in(1.1.3) if there exists ε > 0 such that for any setF ⊂ R n such that |F | = |E| and |E∆F | < ε, E α,β,A (E) ≤ E α,β,A (F ). > 0 such that if m < m * , then B[m]is a L 1 -local minimizer in (1.1.2), and if m > m * then B[m]

	The second result deals with local minimality of balls.
	Terminology 1.3.2.

In the case of problem (1.1.2), we know from

[START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] Theorem 1.5

] that there exists a m *

  for any Borel set E. At this stage we have two sets S * and S * dened by (1.3.1) and (1.3.2), such that if ∈ S * then B[m] is a L 1 -local minimizer in (1.1.3), and if m

	m |B|	1/n	|B|	1/n	/ ∈ S * then B[m] is
	not a L 1 -local minimizer in (1.1.3). We are left to study the variations of the functions

Thus we show that F k is a Λ-minmizer for some Λ uniform in k, and that |F k ∆B| -→ k→∞ 0, which implies by classical regularity theory that F k is an almost spherical set. Up to translating and rescaling F k we can apply inequality

(1.3.3)

. Only simple manipulations are left to get a contradiction.

  1.4.1 Hausdor convergence of large volume minimizers for α < βHere we prove Theorem 1.1.1, i.e. that large volume minimizers of (1.1.3) are almost balls if α < β. Note that if β < 1, we know that large volume minimizers are not exactly balls. Indeed, in virtue of Proposition 1.3.3, balls are not even local minimizers in this case.

  According to Lemma 1.4.1, if γ has been taken large enough, we can assume that u ≤ ε, and so µ n+β ≤ 1+ n+β n + η u. Then, using P

  From lemma 1.4.1 we know that if γ has been taken large enough we have |F | < |B| ε

	2	n	. Thus we can apply lemma 1.4.4 below with r 0

  1.4.26) Remark that by Theorem 1.1.1, the condition u L ∞ (∂B) < ε is satised if γ has been taken large enough. The inequality (1.4.26) will result from this computational lemma, whose proofis postponed : Lemma 1.4.7. Given a measurable function u : ∂B → R with u L ∞ (∂B) < 1, set for t ≥ 0 E Assume that |E t | = |B|.Then, for t small enough, depending only on the dimension n, we haveU β,A (E t ) ≥ U β,A (B) + Aβ t 2 2 u 2 L 2 (∂B) -C(n, β)t 3 u 2 L 2 (∂B) ,

					(1.4.27)
	and
		V α (E t ) ≥ V α (B) -	t 2 2	[u] 2 1-α 2	-α(n + α) u 2 L 2 (∂B)
			-C(n)t 3 [u] 2 1-α 2	+ αV α (B) u 2 L 2 (∂B) ,	(1.4.28)
	where	[u] 2 1-α	=	|u(x) -u(y)| 2
		2	∂B×∂B

t := {s(1 + tu(x))x, x ∈ ∂B, s ∈ [0, 1)}.

  Recalling equation (2.2.3) and denition (2.2.4), we have

	T →∞	δ T ≥ 0.

  From this and (2.2.15) we deduce lim infT →∞ δ T ≥ (2π) n R -α m,and so letting R go to ∞ we get lim inf T →∞ δ T = 0, and so (2.2.9) is proved. Putting (2.2.7), (2.2.8) and (2.2.9) together concludes the proof of (2.2.6).

This gives a contradiction if k is large enough. Thus,(1.3.14) is true. This concludes the proof.

This gives a contradiction if k is large enough. This concludes the proof.

Recall that R > 1, so that if η has been taken small enough, we get that for γ large enough, E α,β,A,γ (F ) ≤ E α,β,A,γ (E γ ), with equality if and only if u=0, i.e. E γ ⊂ B R up to a set of measure 0.

Proceeding to simple asymptotic expansions, we nd that for k large enough we have min f k > 0.We conclude as in case one.

Remerciements

In particular it is clear that if the geodesic γ is long enough (i.e (b -a) is large enough), then we have c(γ, λ, u m ) ≥ 0. Hence we can assume that the inmum in (3.5.2) is taken over unit speed geodesics whose length is bounded by a given constant. By continuity of c for the uniform convergence of curves, and compactness (for the uniform convergence of curves) of geodesics with bounded length in the compact manifold M , the inmum in (3.5.2) is a minimum, attained at a point (γ, λ), with γ dened on a segment [ Recalling that u m ≤ d b , we get c(γ, µ, u m ) ≥ c(γ, µ, d b ) > 0, (3.5.4) where the last inequality comes from the C d -semiconcavity of d b . In particular, µ = λ. Let us assume for instance that µ < λ (the case µ > λ beeing similar) and show that c(γ, λ, u m ) is not minimal in (3.5.2). Figure 3.3 may help justify intuitively the following construction. Let ν ∈ (0, 1) be such that ν µ ab b = λ ab .

and so (3.5.9) implies (3.5.8). Now to prove (3.5.9), the strategy is to show that it holds on the boundary of E m , and to apply the maximum principle to a suitable superharmonic function. If the surface M was at, as we will see in the following, we could apply this strategy directly to det D 2 u m . Let us rst compute its Laplacian. Given any local orthonormal basis (e 1 , e 2 ) of the tangent plane to M , we set D 2 ij u m := D 2 u m (e i , e j ).

(3.5.10)

Now this right hand side can be conveniently computed in a coordinate-free way using classical tensorial dierential calculus on Riemannian manifolds with abstract index notation. We write down the explicit computations here as it does not seem to be a classical identity. The metric tensor of M is denoted by g ab and the Riemann curvature tensor by R abcd . We will denote by |T | the norm of any tensor T . We have

We will use the notation

In the following, any constant that depends on M only will be denoted by C M . By denition of the Riemann tensor we have

and so

(3.5.12)

Likewise,

Chapter 4

Numerical computation of the cut locus

Introduction

Let S be a compact real analytic surface without boundary embedded in R 3 , and b ∈ S any point of S (that can be thought of as a base point).

Denition 4.1.1. The cut locus of b in S can be dened as the closure of the set of points p ∈ S such that there exists at least two minimizing geodesics between p and b. We will denote it by Cut b (S). Equivalently, it is also the set of points around which the distance function to the point b -denoted by d b -is not smooth.

We refer to Chapter 2 for a more detailed introduction to the notion of cut locus. The cut locus is a fundamental object in Riemannian geometry, and it is a natural problem to try and nd ways to compute it numerically. In this chapter, we propose a numerical approximation of Cut b (S), based on the convex variational problem on S (3.1.1), with proven convergence see Section 4.5 for some numerical results. It is not trivial to compute Cut b (S) because it is not stable with respect to C 1 -small variations of S, see for instance [START_REF] Albano | On the stability of the cut locus[END_REF]Example 2]. In particular, one cannot approximate the cut locus of S with the cut locus of a piecewise linear approximation of S.

Related works.

Let us review the techniques used in the past by dierent authors to approximate the cut locus. We may divide them into two categories.

Geodesic approximation on parametrized surfaces. This approach was used in [START_REF] Sinclair | Loki: Software for computing cut loci[END_REF] and [START_REF] Misztal | Cut locus construction using deformable simplicial complexes[END_REF].

In [START_REF] Sinclair | Loki: Software for computing cut loci[END_REF], on genus 1 parametrized surfaces, the authors computed a degree 4 polynomial approximation of the exponential map using the geodesic equation, and deduced an approximation of the cut locus from there. In [START_REF] Misztal | Cut locus construction using deformable simplicial complexes[END_REF], the authors used the deformable simplicial complexes (DSC) method and nite dierences techniques for geodesic computations, to compute geodesic circles of increasing radius and their se-intersection, i.e. the cut locus. They apply the method to genus 1 surfaces. These papers contain no proof of convergence of the computed cut locus.

Exact geodesic computation on discretized surfaces. This approach was used in [START_REF] Itoh | Thaw: A tool for approximating cut loci on a triangulation of a surface[END_REF] and [START_REF] Dey | Cut locus and topology from surface point data[END_REF].

In [START_REF] Itoh | Thaw: A tool for approximating cut loci on a triangulation of a surface[END_REF], the authors computed the geodesics on a convex triangulated surface. They deduced an approximation of the cut locus of the triangulated surface, and ltered it according to the angle formed by the geodesics meeting at a point of the approximated cut locus, to make their approximation stable. They applied the method to ellipsoids. There is no proof of convergence.

Appendix A Appendices to part I A.1 Semiconcavity

Some of the properties of semiconcave functions we need are given in [START_REF] Petrunin | Semiconcave functions in Alexandrov's geometry[END_REF], in the framework of Alexandrov spaces, and sometimes without proof. A more detailed reference for semiconcave functions in the framework of euclidean spaces is [START_REF] Cannarsa | Semiconcave functions, Hamilton-Jacobi equations, and optimal control[END_REF]. In this appendix, we give complete proofs of the properties of semiconcave functions that we need in the framework of smooth Riemannian manifolds. To our knowledge, Proposition A.1.4 is not proved (or even actually stated) anywhere in the literature. We could not nd a proof of the formula for the directional derivatives of distance functions (Proposition A.1.10) in the Riemannian context either. Here, M is any smooth Riemannian manifold without boundary, and g its metric.

For any a, b ∈ R and λ ∈ [0, 1], we dene λ ab := (1 -λ)a + λb. is convex. This is equivalent to having, for any λ ∈ [0, 1],

We say that u is semiconcave if u is C-semiconcave for some constant C > 0. We say that u is locally semiconcave if for any p ∈ M , u is semiconcave in a neighborhood of p.

is convex.

We will need the following lemma to estimate the dierence between two geodesics linking a pair of given points, for two dierent metrics.

Lemma A.1.3. Let g be a metric on the unit ball B(0, 1) ⊂ R n . There exists a constant B > 0 such that for any unit speed geodesic γ : [a, b] → (B(0, 1), g) and λ ∈ [0, 1], we have

and so f is semiconcave on V , as a function of R n . Reciprocally, let us assume that u • ψ -1 is locally semiconcave as a function of R n for any chart ψ. Then, we can show that u • ψ -1 is locally semiconcave for the metric ψ g, for any chart ψ, by using the same technique. From there we deduce that u is locally semiconcave. This concludes the proof.

Remark A.1.5. One can show that any semiconcave function on M is also Lipschitz, using the analogous property for functions on the real line. Therefore, the Lipschitzianity assumption is not actually needed.

By composition with charts, semiconcave functions inherits some properties of concave functions. More precicely, let ψ : B(p, r) → B(0, r) ⊂ R n be a normal coordinate chart at p, for some r > 0. The function u • ψ -1 is semiconcave as a function of R n . Furthermore, the dierential of ψ at p is an isometry. The following proposition then follows from the properties of concave functions on R n . Proposition and denition A.1.6. Let u : M → R be a locally Lipschitz and semiconcave function. At every point p ∈ M , u admits a directional derivative

where γ : [0, 1] → M is any curve such that γ(0) = p and γ(0) = v. Moreover, the map v → ∂ + v u(p) is concave and positively homogeneous on T p M . In particular, if it attains a positive maximum in the closed unit ball of T p M , then this maximum is attained at a unique vector v p . Otherwise, we set v p = 0. We dene the generalized gradient of u at p by

The norm of ∇u(p) is given by the following formula:

Proposition A.1.7. Let u : M → R be a locally Lipschitz and locally semiconcave function. Let x, y ∈ M be such that there exists a geodesic from x to y (which is always the case if M is complete). Then,

The estimate follows.

Proposition A.1.8. Let C > 0. If (u k ) is a sequence of locally Lipschitz C-semiconcave functions that converges pointwise to a function u ∞ , then u ∞ is also semiconcave, and for any sequence of points (p k ) that converges to a point p ∞ , we have

This proves

Now, for every t > 0, let γ t ∈ geod(exp p (tv), K). For t small enough, the length of γ t is bounded by d K (p) + 1. By compactness of the set of geodesics of length bounded by a given constant, there exists a sequence of positive numbers (t n ) n≥0 that converges to 0, such that γ n := γ tn converges to a unit speed geodesic γ as n → +∞. As K is closed, γ is a geodesic from p to K. Moreover, we have

, where inj(M ) is the injectivity radius of M . In particular for any (x, y) such that d(x, y) < R and x = y, the distance function d(

For n large enough, we have exp p (t n v) ∈ U and b n ∈ V , and so

where ∇ 2 is the gradient with respect to the second coordinate. We have

In particular,

With (A.1.1), this concludes the proof.

If T = T ab , we get T 2 = T ab T ab .

Tensor elds. A tensor eld T on M is the collection of tensors of T p M for every point p ∈ M . A tensor eld T is said to be smooth if its coordinates in a smooth basis of T M are smooth.

Covariant derivative. Let X(M ) denote the set of smooth vector elds over M . The covariant derivative D of M (or Levi-Civita connection) is the unique map

that is C ∞ (M )-linear in the rst variable, R-linear in the second variable and such that for

(compatibility with the metric)

where

is a well dened endomorphism on every tangent space T p M , i.e a (1, 1) tensor eld. Therefore we can dene the (total) covariant derivative tensor DX of X by the relation

In abstract index notation, the tensor DX will be denoted by

The covariant derivative can be extended to any kind of tensor, by requiring only that it satises some product rules. Therefore, for any tensor (k, l) tensor T , there is a well dened (k, l + 1) tensor DT . The second covariant derivative (or Hessian) of T is D 2 T = D(DT ).

In abstract index notation, the covariant derivative of a tensor eld T a 1 ... The covariant derivative saties the following product rule :

The covariant derivative commutes with the trace. In particular, an expression of the type D a T bc bc makes sense, as we do not need to specify if it denotes the derivative of the trace of the tensor T , or the trace of the derivative of the tensor T . Let us give an example of computation with the covariant derivative on tensors. Using the facts that the covariant derivatives commute with the trace and that it satises a product rule on tensor products, we deduce for instance that for any tensor H ab ,

By this example, we hope to show that computations with covariant derivatives in abstract index notation are fairly intuitive.

Warning. Let T be any tensor eld, and X and Y some vector leds. The tensor D a D b T X a Y b is not the same as D X (D Y T ). Ideas of the proof. The method used in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] still applies to our functional E α,β,A = P + V α + U β,A . Given m > 0, let us proceed to a rescaling of the functional and set

Thus we are reduced to nding the values of γ > 0 such that the unit ball B centered at 0 is a local minimizer of E α,β,A,γ . Following [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF]Section 6] we can compute the second variation of E α,β,A,γ at B. The terms P and V α are treated in [START_REF] Figalli | Isoperimetry and stability properties of balls with respect to nonlocal energies[END_REF] and the term U β,A adds no further diculty. We nd that given any smooth compactly supported vector eld X, such that the volume of B is preserved under the ow (Φ X t ) t>0 of X, we have:

• ν B is the unit outer normal vector to ∂B,

• a k (X • ν B ) are the coecient of the function X • ν B : ∂B → R with respect to an orthonormal basis of spherical harmonics,

The theorem will then result from the isoperimetric inequality: P (E) > P (B) if E = B. We divide the proof of (1.4.15) into two steps. In step one we compare E to the subgraph of a function over the sphere, by concentrating the mass of E on each half line through the origin.

In step two, we show that (1.4.15) holds for subgraphs of suciently small functions over the sphere.

Step one. For any x ∈ ∂B, dene u(x) ∈ R by the equation

Then, set

(1.4.17)

We have

thus E u satises the volume constraint. Now we estimate the variation of U β,A . From Theorem 1.1.1 we know that, taking γ large enough, we can assume B 1 2 ⊂ E. Thus we have

where we have set

Here we need a simple lemma from optimal transportation on the real line.

Lemma 1.4.6. Given a measurable set S ⊂ (1/2, ∞) such that S r n-1 dr < ∞, let u > 0 be such that S r n-1 dr =

r n-1 dr.

Then, there exists a unique non-decreasing measurable map T : (1/2, u) → (1/2, ∞) such that

Chapter 2

Numerical minimization

In this Chapter we present our method and results for the numerical minimization of the variational problem (1.1.3). We apply this method to the original problem (1.1.2). Indeed, the theoretical knowledge we have so far on problem (1.1.2) raises two natural questions. Is it always the case (i.e. for any value of α ∈ (0, n)) that there is no minimizer for m large enough? Is there a set of parameters n, α and m, such that there exists a minimizer that is dierent from a ball? Our numerical results indicate that in dimension 2, the answers are positive and negative respectively.

Preliminaries

Γ-convergence

In this section, we introduce the notion of Γ-convergence, which we will be using to justify our numerical method. This is the notion of convergence of functionals one generally wants to use when studying variational problems. We refer to the book [START_REF] Braides | Gamma-convergence for Beginners[END_REF] for more details about Γ-convergence. The lecture [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF] also contains a brief introduction to Γ-convergence that is sucient to our needs.

Denition 2.1.1. Let X be a topological space, F a real functional on X, and (F ε ) ε>0 a family of real functionals on X. We say that F ε Γ-converges to F as ε goes to 0, and we write

F, if the two following assertions hold:

1. for any u ∈ X and any family

2. for any u ∈ X, there exists a family (u ε ) ε>0 of X such that u ε -→ ε→0 u and

When dealing with the perimeter term, we will also use the following relaxation of the Γ-convergence denition. This denition and notation is not standard. F, if the assertion 1 of the previous denition holds for any u ∈ X, and the assertion 2 holds for at least one u ∈ X that minimizes the functional F.

Remark 2.1.3. We will actually use the following equivalent two assertions

• for any u ∈ X and any family (u ε ) ε>0 of X such that u ε -→ ε→0 u, we have

• for any u ∈ X (respectively, in the case of Γ-convergence, there exists a minmizer u ∈ X of F such that), there exists a family (u ε ) ε>0 of X such that u ε -→ ε→0 u and lim sup ε→0

Remark 2.1.4. We also say that a sequence of functionals (F N ) Γ-converges to a function F if the assertions above hold with ε and 0 replaced with N and +∞.

Both denitions guarantee that if, for any ε > 0, u ε is a minimizer of F ε in X, and the family (u ε ) converges to u ∈ X as ε → 0, then u is also a minimizer of F in X. To minimize numerically a given functional F, our strategy is the following:

1. nd a sequence of discrete functionals (F N ) N ∈N that Γ-converges to F, 2. nd numerically a minimizer u N of F N for some N considered 'large', 3. extrapolate a limit point u of the sequence (u N ) N ∈N from the results. This u should be a minimizer of F.

For point 3 to even have a chance to work, we need the sequence (u N ) N ∈N to be precompact: Terminology 2.1.5. Following [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF], we say that a family of functionals (F ε ) ε>0 dened on a metric space X enjoys property (C) (for compactness) if for any family (u ε ) ε>0 of elements of X such that (F ε (u ε )) ε>0 is bounded, there is a subsequence of (u ε ) ε>0 that converges in X.

If a family of functionals (F ε ) ε>0 enjoys property (C) and Γ-converges to a limit functional F when ε goes to 0, then we know that for ε small enough, minimizers of F ε are close to minimizers of F.

To conclude this section, we have the following elementary proposition.

Proposition 2.1.6. If G : X → R is continuous and

Modica-Mortola theorem

The numerical treatment of the perimeter is classical. To deal with the perimeter term, we will use the well-known Modica-Mortola theorem. This theorem allows us to relax the perimeter functional on sets, i.e. charateristic functions, into a functional on functions taking values in [0, 1]. Therefore we will be able to use the vector space structure of functions and, after disctretization (step three), usual optimization tools for functionals on R d . W (u)du. We dene the set X, and the functionals F ε : X → R by

and

Moreover, if E ⊂⊂ Ω is a compact subset of Ω of nite perimeter, then there exists a family of functions (u ε ) ε>0 in X, such that u ε -→ ε→0 1 E , and

Finally, the family of functionals (F ε ) enjoys property (C).

Reader-friendly proofs of these assertions can be found in [START_REF] Alberti | Variational models for phase transitions, an approach via Γ-convergence[END_REF]. Here, we will only give the following heuristic. If ε > 0 is very small, then ε -1 W (u) is very large, except if u is 0 or 1. Therefore the minimizers of F ε are functions that are equal to 0 or 1 except on a set of small measure.

Method of the numerical minimization

We present a series of three modications of the variational problem (1.1.3) to arrive at a nite dimensional variational problem that can be easily numerically solved. All steps are justied by a Γ-convergence and compactness result.

Step one is standard when dealing with the perimeter. We use the classical Modica-Mortola theorem to relax the functional on sets, i.e. charateristic functions, into a functional on functions taking values in [0,[START_REF] Adelstein | Morse theory for the uniform energy[END_REF]. This allows us to use the vector space structure of functions and, after disctretization (step three), usual optimization tools for functionals on R d .

Step two is the key step for dealing with the non-local term V α . We replace the ambient space R n with a large square with periodic boundary conditions, whose size is a new relaxation parameter. Then, we can approximate the non-local term V α by a simple expression in Fourier variable.

In step three, we discretize the problem by considering only trigonometric functions with frequencies lower than some integer N , and by computing the integral terms with riemann sums.

Let us now describe and justify each step precisely.

Step one. We use the classical Modica-Mortola theorem to replace this problem on subsets of R n , i.e. functions taking only values 0 or 1, with a problem on functions taking any value between 0 and 1. Let m > 0. First, remember that we proved in the proof of Proposition 1.2.1 that minimizers of our problem (1.1.3) are uniformly bounded. Let Ω be a Lipschitz bounded open set such that one minimizer of (1.1.3) is compactly included in Ω. Given a (small) ε > 0, we dene the set X, and the functionals F ε : X → R and F : X → R by

and

where we have used the natural notation V α (u) =

|x-y| n-α dxdy, and W is the following double well potential on [0, 1]: W (x) = x(1 -x). Then, from the Modica-Mortola theorem and the fact that the two last terms of the functionals F ε and F are continuous on X, we have the following proposition. Proposition 2.2.1. The following Γ-convergence holds on X, endowed with the strong L 1 (R n )topology.

Moreover, (F ε ) enjoys property (C).

Step two. For simplicity, we will rst reduce the domain of denition of F ε , and use a stronger topology in this step. Let us set

endowed with the weak H 1 (R n )-topology. The non-local repulsive term has a simple expression in Fourier variable :

with û the Fourier transform of u and C(n, α) :=

, and Γ the usual gamma function.

This can be seen by noting that V α (u) = uI α (u) with I α (u) the Riesz potential of u, and using the Fourier expression of the Riesz potential (see [START_REF] Stein | Singular integrals and dierentiability properties of functions (PMS-30)[END_REF]Part V]). To exploit this formula, we will replace the ambient space R n with a large torus of size T > 0. Thus we will approximate

where c k,T (u

for some (large) T > 0. More precisely, let us dene the functional F ε,T : X → R by

Then, we have the following proposition.

Proposition 2.2.2. The following Γ-convergence holds on X , endowed with the weak H 1 (R n )topology:

Moreover, (F ε,T ) T >0 enjoys property (C).

We emphasize the following remark:

Remark 2.2.3. To prove property (C) in this step, the assumption that all functions are supported in a given bounded set Ω is important. Numerically, we also observe that the method fails without this assumption. See Section 2.3 for further comments.

Let us prove Proposition 2.2.2.

Proof of Proposition 2.2.2. Proof of property (C). Let (u T ) T >0 be a sequence of X such that (F ε,T (u T )) T >0 is bounded.

Then, (u T ) T >0 is bounded in H 1 (R n ), so up to taking a subsequence, it converges weakly to a function u ∈ H 1 (R n ). Up to taking another subsequence, we may also assume that the convergence hold almost everywhere. So, u takes its values in [0, 1], and we have u |Ω c = 0. In particular, we also have u |Ω ∈ H 1 0 (Ω). The condition u = m is veried as well by dominated convergence. Hence property (C) is veried.

Proof of the lim inf inequality for Γ-convergence. Here we assume that (u T ) T >0 is a sequence of X that converges to a limit u ∈ X with respect to the weak H 1 (R n )-topology. We need to show that lim inf

Because of the weak H 1 (R n )-convergence, we have lim inf

(2.2.7) By Fatou's lemma, we also have lim inf

(2.2.8) endowed with the weak H 1 ((-T /2, T /2) n )-topology. Let us assume that T is large enough so that Ω ⊂⊂ (-T /2, T /2) n . If u ∈ X takes its values in [0, 1] and veries u ∈ H 1 0 (Ω) and u | Ω c = 0, we may extend it by 0 outside of (-T /2, T /2) n , to get a function u ∈ X , and dene F ε,T (u) := F ε,T ( u). Otherwise, we set F ε,T (u) = +∞. This way, the functional F ε,T is well dened on X . For N ∈ 2N * large, we consider the space

and

Then, we dene the functional F ε,T,N :

Then, we have the following proposition. Proposition 2.2.4. The following Γ-convergence holds on X , endowed with the weak H 1 ((-T /2, T /2) n )topology:

Moreover, (F ε,T,N ) N >0 enjoys property (C).

In the proof of 2.2.4, we will use the following technical lemma, which shows that a triogonometric function whose frequencies are lower than N is well represented by its values on a grid with step size 1/N . Lemma 2.2.5. Let (u N ) be a converging sequence in

Then, for any bounded uniformly continuous functions φ : R → R and ψ : [0, 1] n → R, we have

Equivalently, for any T > 0, let (u N ) be a converging sequence in

Then, for any bounded uniformly continuous functions φ : R → R and ψ : [-T /2, T /2] n → R, we have

Proof. We only need to prove the lemma in the case of functions on [0

We also dene

where the (c k (u N )) k∈{-N 2 +1,..., N 2 } n are the Fourier coecients of u N . Then, we can rewrite

Let (K N ) be a sequence of positive integers, diverging slowly enough (to be precised later) to +∞. We dene some functions v N and v N by

Also, noting that the family (e

The function φ being uniformly continuous and bounded, it is easy to show that (2.2.19) is equivalent to

which is in turn equivalent to

(2.2.20)

Now, we have

where we have used the Hölder inequality in (2.2.21), and C(n) denotes a constant depending on n only. From this last estimate, we get

Denoting by ω φ the modulus of continuity of φ, this yields

0, and recalling that the sequence Proof of the lim inf inequality. Let (u N ) be a sequence of X that converges to a function u ∈ X , and such that for every N ∈ 2N * , u N ∈ E N . We need to show that lim inf

We may assume that the left hand-side is nite. Up to taking a subsequence, we may then assume that (F ε,T,N (u N )) N ∈2N * is bounded. Because of property (C), up to taking another subsequence, we may assume that (u N ) converges strongly in L 2 ((-T /2, T /2) n ) and almost everywhere to u.

Let us show that u takes its values in the interval [0, 1], and that u | Ω c = 0.

To prove that u takes its values in [0, 1], let us argue by contradiction and assume it is not the case. Then, there exists a continuous function φ : R → R + that is compactly supported in the complement of [0, 1], and such that

Applying Lemma 2.2.5 with ψ = 1, we nd

By the dominated convergence theorem, this can be rewritten as

which gives a contradiction. So u takes its values in [0, 1].

To prove that u | Ω c = 0, we also argue by contradiction, and assume it is not the case. Then, there exists a continuous function ψ : [-T /2, T /2] n → R that is supported in Ω c , and such that

Applying Lemma 2.2.5 with φ(x) = x, we nd

By the dominated convergence theorem, this can be rewritten as

which gives a contradiction. So u | Ω c = 0.

By the weak H 1 ((-T /2, T /2) n ) convergence, we have lim inf

By the strong L 2 ((-T /2, T /2) n ) convergence, we have lim

Thus, the only problematic terms are W N and U β,A,N . Let us apply Lemma 2.2.5 to (u N ), with ψ = 1 and φ = W , where W is any uniformly continuous bounded extension of W . We get

(2.2.22)

By the dominated convergence theorem, we have lim

With (2.2.22), this implies lim

Using the same stretegy, we may also show that lim

Proof of the lim sup inequality. Let u ∈ X . We need to nd a recovery sequence (u N )

such that for any N ∈ 2N * , u N ∈ E N , and lim sup

The domain of niteness of F ε,T is

The set C ∞ ∩ D F ε,T of smooth functions of D F ε,T is a dense subset of D F ε,T for the strong H 1 ((-T /2, T /2) n )-topology. Moreover, the functional F ε,T is continuous over D F ε,T for the strong H 1 ((-T /2, T /2) n )-topology. Therefore, we may assume that u ∈ C ∞ ∩ D F ε,T . Let (c k ) k∈Z n be the Fourier coecients of u, so that

The sequence (v N ) cannot be used directly as a recovery sequence because it does not verify

However, because u is smooth, by classical uniform estimates for the Fourier serie of a C n+4 function, there exists a constant C > 0 such that for any

We recall that the discrete Fourier transform is a bijection: for any two familys of complex numbers (w

Note that, because of (2.2.24), we have the estimate

Then, by the bijectivity of the discrete Fourier transform, we have w N ( jT N ) = w j . Now, w N veries the constraints (2.2.23), but it may not verify w N = m. However, we have

We will modify the values w N ( jT N ) on a small subset of Ω, so that the new values still verify (2.2.23), but the mean of the new function is m. Recall that u is continuous, null outside Ω, and not identically null. In particular, there exists η > 0 and x ∈ Ω such that η ≤ u ≤ 1 -η on B(x, η). Because of (2.2.24) and (2.2.26), we may also require, up to taking η smaller, that for any N large enough,

(2.2.28)

Let K be the cardinal of the set

For N large enough, there exists j ∈ {-N 2 + 1, . . . , N 2 } n such that jT N ∈ B(x, η), so that K = 0.

So we can set

Note that, because of (2.2.27) and (2.2.28), for some constant C > 0, we have for any N large enough,

Finally, let us dene

Then, by the bijectivity of the discrete Fourier transform, we have

and so by construction u N veries the constraints (2.2.23). Moreover, we have

0.

So (u N ) converges strongly to u in H 1 ((-T /2, T /2) n ). We can now conclude, using Lemma 2.2.5 as in the proof of the lim inf inequality, that lim N →+∞ F ε,T,N (u N ) = F ε,T (u).

This concludes the proof.

Numerical results

In the physical case, i.e. in dimension n = 3 and for α = 2, R. Choksi and M. Peletier conjectured the following (see [START_REF] Choksi | Small volume-fraction limit of the diblock copolymer problem: Ii. diuse-interface functional[END_REF]Conjecture 6.1]):

Conjecture 2.3.1. For n = 3 and α = 2, as long as there is a minimizer in Gamow's liquid drop problem (1.1.2), it is a ball. Also, when there is no minimizer, the inmum of the energy is attained by a nite number of balls of the same volume, innitely far away from each other.

As far as it is known, this conjecture may hold for any n and α. In any dimension n ≥ 2, for α close enough to n, M. Bonacini and R. Cristoferi proved that as long as a minimizer exists for problem (1.1.2), it is a ball. Moreover, when there is no minimizer, the showed that • for T = 5π: Here the box Ω in which all functions are supported (see Section 2.2) has been chosen to be a square of diagonal length π (and is represented by white lines). We emphasize that this box is needed to get the right minimizers, both theoretically and numerically. Theoretically, the condition that functions are supported in a xed bounded set is needed for the compactness property (C) (again see Section 2.2) to be satised, both in step one and in step two, as we let the size of the domain T go to innity. Numerically, without this box, for c m = 1.5, simulations yield two balls (instead of one as shown on Figure 2.1a) that get further and further away from each other as T increases. But this conguration does not converge to an admissible candidate, so it denitely does not converge to a minimizer.

We observe that for c m = 1.6, we get two balls in opposite corners of the square Ω: it is consistent with the expected repulsive behaviour of the non-local term V α . Moreover, using (2.3.1) and (2.3.2), we nd that, if conjecture 2.3.2 is true, there must be a minimizer up to c m 1.67. Numerically, we nd that there is a minimizer up to a constant c m ∈ (1.5, 1.6), which is relatively close to 1.67. We also observe similar results for dierent values of α, including in the near eld-dominated regime α < 1.

For Ω a disk of diameter π, if one increases further c m , we get three balls located near the boundary of Ω, as shown in Figure 2.2a for c m = 3.0. This is consistent with the conjecture that the energy is minimized by balls of the same volume. To illustrate the eect of the conning potential, we display in Figure 2.2b the minimizer for c m = 3.0, A = 1 and β = 16.

Finally, let us mention that the number of discretization points is N = 2 11 in each direction.

Numerical minimization is made using the solver IPOPT [START_REF] Wächter | On the implementation of a primal-dual interior point lter line search algorithm for large-scale nonlinear programming[END_REF]. The computation time on a standard computer is about an hour.