Benedikt Wirth

Edouard Santambrogio

Blanche Oudet

Bouet

Grazie Alle

Parigi Tommaso

Marta E Gregorio E Mar- Tino

Francesco Tommaso

Matteo Filippo

Pietro Perch Tornare

Marta Hlose

Lucia Caterina

Clelia Paince

Otto Roby

Laura

Daniela Maria

Andrea Leo

Elena Maria

Sara Pierluca

Pietro Chiara

Martino Grazie

" Maddalena

Elisa Silvia

passati e presenti

First of all, I would like to express my deep gratitude to my two supervisors, Antonin Chambolle and Benoit Merlet, for their guidance. I will always be grateful for the suggestions on the subjects and I am grateful to Prof. Chambolle for his inspirations. I will always remember the first time he introduced to me the theory of Gamma convergence and the subject of branched transportation. Seeing him work at a blackboard is something really fascinating.

Introduction

When designing a supply-demand distribution network it is convenient to give it a tree structure in which it is preferable to regroup mass in the transportation process. This assumption emerges from numerous observations, for instance the structure of the blood vessels in the cardiovascular system is required to distribute blood from a concentrated source in the heart to a widespread volume or vice-versa, the root system of a tree needs to recollect water from the soil. In these situations we may observe how broad and long vessels are preferable rather than thin spread out ones. The assumption we make is that the actual observed network is optimal with respect to some given cost among all possible networks developing from a source and irrigating a given sink. These structures appear in a wide range of situations (figure 1) and many efforts have been made by the mathematical community in order to give a precise model able to describe all the observable features of these networks.

Figure 1: On the left: root network of a tree. On the right: angiography of an eye in which it is possible to recognize the tree structure of the network of blood vessels.

A first well known approach in the framework of graph theory was proposed by Gilbert in [START_REF] Paolini | An example of an infinite Steiner tree connecting an uncountable set[END_REF] where he deals with the Steiner Minimal Tree [START_REF] Ambrosio | Topics on analysis in metric spaces[END_REF][START_REF] Paolini | Existence and regularity results for the Steiner problem[END_REF] problem. The latter consists in finding the graph connecting a given set of points {x 0 , . . . , x N } with minimal total length. More formally, a Steiner minimal tree is the solution of the variational problem argmin H 1 (K) : K compact, connected and contains x 0 , . . . , x N ,

where H 1 (K) is the Hausdorff 1-dimensional measure of F (the length of K, if it is 1-dimensional and sufficiently smooth). As stated in Courant and Robbins [START_REF] Courant | What is mathematics?[END_REF] the Steiner minimal tree problem can be thought of as a naif model for the network of highways connecting a set of cities. The drawback of the model is that the local intensity of the traffic is not taken into account. Nevertheless it allows to appreciate the issues emerging from these models. As observed in the quoted paper [START_REF] Paolini | An example of an infinite Steiner tree connecting an uncountable set[END_REF] in a Steiner minimal tree, differently from the Minimal Spanning Tree [START_REF] Kruskal | On the shortest spanning subtree of a graph and the traveling salesman problem[END_REF], new vertices

Introduction

Figure 2: Steiner Minimal Tree connecting 10000 points randomly distributed in the plane. The problem was solved using the GeoSteiner algorithm [START_REF] Winter | Euclidean steiner minimum trees: An improved exact algorithm[END_REF], which is currently the most efficient exact algorithm for computing minimum Steiner trees. may be added in order to minimize the total length thus, rather than the network itself, the real unknown is its topology. An example of this situation is shown in Figure 3. This feature appears as well in other models in which the cost per unit length depends on the intensity of the traffic flux [START_REF] Gilbert | Minimum cost communication networks[END_REF]. In light of this high combinatorial complexity, the problem is in the list of NP-complete problems from Karp [START_REF] Karp | Reducibility among combinatorial problems[END_REF] and it is still an active field of research even in the operational research community [START_REF] Forte | Iterated local search algorithms for the euclidean steiner tree problem in n dimensions[END_REF].

(1, 0)

(-1/2, √ 3/2) (-1/2, - √ 3/2) (1, 0) (-1/2, √ 3/2) (-1/2, - √ 3/2)
Figure 3: On the left: Minimal Spanning Tree connecting three points situated at the vertices of an equilateral triangle (length = 2 √ 3). On the right: Steiner Minimal Tree constrained to connect the same set of points (length = 3). In dark blue the additional vertex which allows to decrease the total length. The purpose of this thesis is to devise approximations of some Branched Transportation problems. Branched Transportation is a mathematical framework for modeling supply-demand distribution networks which is more general than the Steiner problem presented above. In particular the supply factories and the demand locations are Introduction modeled as measures supported on points and the network is interpreted as a vector measure, eventally the problem is cast as a constrained optimization problem. Given a function h, the transport cost of a mass m along an edge with length is h(m) and the total cost of a network is defined as the sum of the contributions of all its edges. The branched transportation case corresponds to the specific choice h(m) = |m| α with α ∈ [0, 1). The sub-additivity of the cost function, h(m 1 + m 2 ) ≤ h(m 1 ) + h(m 2 ), ensures that transporting two masses jointly is cheaper than doing it separately. This formulation shares much of the numerical complexities presented above in the case of the Steiner Minimal tree problem. In this work we introduce various variational approximations by means of elliptic-type functionals to obtain more efficient numerical schemes. Eventually the proposed method is generalized to Plateau-type problems, which is a framework to model soap films spanning a given boundary. In its more general formulation the unknown of this problems is a k-dimensional surface in R n spanning a (k -1)-dimensional boundary and minimizing a certain cost. Branched transportation corresponds to a Plateau type problem for the choice k = 1.

Description of the model

Let us introduce precisely the framework for Branched Transportation [START_REF] Bernot | Optimal transportation networks[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF]. First we introduce transport networks in a open set Ω ∈ R n , and the associated cost functional. For this purpose consider a segment Σ ⊂ Ω, a positive real number m ∈ R + and the vector τ ∈ S n-1 tangent to Σ. The writing

m τ H 1 Σ (2)
defines a vector valued measure, where H 1 Σ is the Hausdorff 1-dimensional measure in R n restricted to the segment Σ. Intuitively, the Radon measure H 1 Σ associates to any measurable set A the length of A ∩ Σ. We say that a vector valued measure σ ∈ M(Ω, R n ) is polyhedral if it is a finite sum of measures of the form (2), namely σ = i m i τ i H 1 Σ i . The action of σ on C 0 (Ω, R n ) is defined by the formula (σ, ϕ) =

Introduction

A transport cost function h : R → [0, +∞) is an application such that h is even, lower semicontinuous, sub-additive, with h(0) = 0.

(3)

Given a transport cost function h we define the Gilbert energy of a polyhedral vector measure σ as

E h (σ) := i h(m i )H 1 (Σ i ).
We endow M(Ω, R n ) with its weak- * topology and extend E h by relaxation, namely for a vector measure σ ∈ M(Ω, R n ) we let E h (σ) := inf lim inf j→+∞ E h (σ j ) : σ j * σ and σ j polyhedral .

By White in [START_REF] White | The deformation theorem for flat chains[END_REF]6] conditions (3) are sufficient in order to extend E h on M(Ω, R n ).

Choosing h(m) = |m| in equation ( 4) we obtain the mass functional which associates to each vector measure σ its total variation

|σ| = sup{(ϕ, σ) : ϕ ∈ C 0 (Ω, R n ), ϕ ∞ ≤ 1}.
Otherwise, with h(m) = χ {m =0} , where χ denotes the characteristic function of a set, E h reduces to the size functional which measures the length of the support of σ, namely σ → H 1 (supp(σ)). Other remarkable choices are represented in Figure 5.

1) m h(m) h(m) = |m| 2) m h(m) h(m) = |m| α 3) m h(m) h(m) = χ {m =0} 4) m h(m) h(m) = (1 + β |m|)χ {m =0} 5) m h(m) h(m) = min{α 0 |m|, α 1 |m| + β 1 }
Figure 5: For h as in the graphs we obtain respectively the: 1) Mass, 2) α-Mass, 3) Size, 4) Affine cost, 5) Urban planning functional.

Introduction

To model the source and the sink of the transport network we introduce two probability measures µ + , µ -∈ P(Ω) and restrict our attention to the vector space X µ + ,µ -⊂ M(Ω, R n ) consisting of those vector measures σ satisfying

div σ = µ + -µ - (5)
in the sense of distributions. As shown in the note [START_REF] Chambolle | Strong approximation in h-mass of rectifiable currents under homological constraint[END_REF] if the relaxation is obtained with respect to polyhedral measures in X µ + ,µ -we still obtain the functional (4).

Finally we are interested in approximating minimizers of the Gilbert energy under the divergence constraint (5), namely:

min {E h (σ) : σ ∈ X µ + ,µ -} . (6)
The Branched Transportation case corresponds to the choice h(m) = |m| α with α ∈ [0, 1) and has been introduced by Xia who has investigated as well the problem of existence and regularity of solutions. In [Xia03] the author, taking advantage of variationals methods, proves the following Theorem 0.1 (Existence Theorem). Given α ∈ (1-1 n , 1] and two probability measures µ + , µ -∈ P(Ω), there exists a vector valued measure σ ∈ X µ + ,µ -for which E h (σ) is minimal. Furthermore for σ ∈ argmin E h (σ) we have the following estimate

E h (σ) ≤ 1 2 1-n(1-α)-1 √ n diam(Ω) 2 .
In a subsequent result [Xia04, Theorem 2.7] the same author addresses the problem of regularity. To state the result we need to introduce the notion of rectifiable vector measure. Namely a vector measure σ is said rectifiable if

σ = m τ H 1 Σ (7)
where Σ, the support of σ as a distribution, is an H 1 -rectifiable set, its H 1 -density is the function m ∈ L 1 (H 1 Σ) and τ : Σ → S n-1 spans for H 1 -a.e. point in Σ the tangent space to Σ. In the following we denote with (m, τ, Σ) the rectifiable measure σ defined in (7).

Introduction

Proposition 0.1 (Generalized Gilbert-Steiner Energy). Let µ + , µ -∈ P(Ω), σ ∈ M(Ω, R n ) with finite total variation and such that div σ = µ +µ -then σ can be decomposed as

σ = σ ⊥ + m τ H 1 Σ
where (m, τ, Σ) is the H 1 -rectifiable component of σ and σ ⊥ is the diffused one. Furthermore

E h (σ) = h (0)|σ ⊥ |(Ω) + Σ h(m) dH 1 . ( 9
)
With an abuse of notation we have denoted h (0) = lim m↓0 h(m)/m.

Before introducing problems involving surfaces and other higher dimensional objects let us highlight the fact that the Steiner minimal tree problem connecting some points {x 0 , . . . , x N } may be modeled in the context of Branched transportation. Firstly, with the choice α = 0, E h reduces to the size functional. Secondly the divergence constraint forces any considered vector measure to join the support of µ + to the support of µ - thus, by choosing µ + = δ x 0 and µ -= 1/N N i=1 δ x i we force x 0 to be connected to each x i . Gathering all together, with these choices, a minimizer σ of ( 6) is supported on a set connecting the points in {x 1 , . . . , x N } to x 0 and has support with minimal total length thus is a solution to (1). The energy introduced above for rectifiable measures supported on 1-dimensional surfaces can be generalized to any dimension k ∈ {1, . . . , n}. To this aim is necessary to introduce the concept of k-currents in R n . Denote with D k (Ω) the space of smooth differential forms on the open set Ω. The vector space of k-currents, D k (Ω), is the dual to D k (Ω) and it is naturally endowed with its weak- * topology. We mainly follow the notation of [START_REF] Krantz | Geometric integration theory. Cornerstones[END_REF][START_REF] Federer | Geometric measure theory[END_REF] the main difference being the use of σ to denote a k-current instead of a latin capital letter. For a current we can define a notion of boundary by duality as follows ∂σ, ω = σ, dω for all (k -1)-differential forms ω.

We call mass of a k-current the supremum of σ, ω among all k-differential forms with comass bounded by 1, and denote it with |σ|. In particular by the Radon-Nikodym theorem we can identify a k-current σ with finite mass with the vector valued measure τ µ σ where µ σ is a finite positive valued measure and τ is a µ σ -measurable map in the set of unitary k-vectors for the mass norm. The relation with vector measure is evident when we consider the fact that the vector spaces Λ 1 R n , Λ 1 R n identify with R n . Thus any vector measure σ ∈ M(Ω, R n ) with finite mass identifies with a 1-current with finite mass and vice-versa. Furthermore the divergence operator acting on measures in the sense of distributions is defined by duality as the boundary operator for currents. Thus, in analogy with what has been presented for vector measures, in equation ( 7), a k-current σ is said to be k-rectifiable if we can associate to it a triplet (θ, τ, Σ) such that σ, ω = values in R + . The vector space of Rectifiable Currents is denoted with R k (Ω). Among these we single out the subset P k (Ω) of rectifiable currents for which Σ is a finite union of polyhedra and θ is constant on each of them, these will be called Polyhedral Chains.

For any k-current σ such that both σ and ∂σ are of finite mass we say that σ is a normal k-current and we write σ ∈ N k (Ω). On the space D k (Ω) we can define the flat norm by

F(σ) = inf {|σ R | + |σ S | : σ = σ R + ∂σ S where σ S ∈ D k+1 (Ω) and σ R ∈ D k (Ω)} ,
which metrizes the weak- * topology on currents on compact subsets of N k (Ω). Finally the flat chains F k (Ω) consist of the closure of P k (Ω) in the F topology. By the scheme of Federer [START_REF] Federer | Geometric measure theory[END_REF]4.1.24] it holds

P k (Ω) ⊂ N k (Ω) ⊂ F k (Ω).
Following the strategy proposed by Fleming [START_REF] Federer | Normal and integral currents[END_REF][START_REF] Fleming | Flat chains over a finite coefficient group[END_REF] in the context of flat chains with coefficients in groups we now define the energy E h on the space of flat chains. Let h be a transport cost function and σ = (m i τ i , Σ i ) a polyhedral current we let

E h (σ) := i h(m i )H k (Σ i ).
In analogy to what has been done before we extend E h on the space of flat chains by relaxation. For σ ∈ F k (Ω), E h (σ) := inf lim inf j→∞ E h (σ j ) : σ j polyhedral and F(σ jσ) → 0 .

In Chapter 3 we look for approximations to problems of the type min{E h (σ) : ∂σ = ∂σ 0 }

where σ 0 is a given polyhedral k-current. These problems have been introduced and studied in [START_REF] Morgan | Size-minimizing rectifiable currents[END_REF][START_REF] Pauw | Size minimization and approximating problems[END_REF] by Morgan, De Pauw and Hardt among others to propose different models for soap film minimal surfaces. The latter is the k-dimensional generalization to the minimization problem defined in (6). As sketched in [START_REF] White | The deformation theorem for flat chains[END_REF][START_REF] White | Rectifiability of flat chains[END_REF] E h has an explicit formulation on rectifiable currents, namely for a rectifiable current (m, τ, Σ) we have

E h (σ) := Σ h(m) dH k .
This result has been proved in [CDRMS17, Proposition 2.6], is and is the consequence of the following polyhedral approximation theorem Theorem 0.3 (Polyhedral approximation). Let h be a transport cost function and let σ = (m, τ, Σ) be a rectifiable k-current. For every δ > 0 there exists a polyhedral k-chain σ = (m i , τ i , Σ i ) such that This result may be seen in correlation with equation (9) presented above. Let us highlight that the polyhedral approximation result from Colombo et al. does not take into account any boundary constraint for the k-currents. An analogous result with boundary constraint has been proved in the note [START_REF] Chambolle | Strong approximation in h-mass of rectifiable currents under homological constraint[END_REF]. We conclude this section with an important sufficient condition for a flat chain to be rectifiable, proved by White in [Whi99a, Corollary 6.1].

F(σ -σ) ≤ δ, i h(m i )H k (Σ i ) ≤ Σ h (
Theorem 0.4 (Rectifiability for currents). Let σ ∈ N k (Ω) be a normal k-current supported on a k-rectifiable set; then σ is rectifiable.

We will take advantage of this theorem even in the context of vector measures. With the notation introduced above it reads as Theorem 0.5 (Rectifiability for vector-valued measures). Let σ ∈ M(Ω, R n ). If |σ|(Ω) + |∇ • σ|(Ω) < ∞, ∇ • σ is at most a countable sum of Dirac masses and there exists a Borel set Σ with H 1 (Σ) < ∞ and σ = σ Σ, then σ is a rectifiable vector measure.

Introduction

The sequence (x ε ) si called recovery sequence for x. Condition (UB) is frequently hard to prove thus it is convenient to find a subset D ⊂ X such that: for every x ∈ X there exists an approximating sequence (x n ) ⊂ D such that x n → x and F (x n ) → F (x). If we are able to recover D then a simple diagonal argument shows that it is enough to verify condition (UB) for all x ∈ D rather than for every x ∈ X. In the context of our work the set D corresponds with the set vector space of polyhedral vector measures. Since the definition of Γ-convergence may appear cumbersome let us provide this alternative characterization that allows to appreciate its relevance in the context of the Calculus of Variations.

Theorem 0.6 (Characterization of Γ-convergence). Let X be a metric space, and for ε > 0 let be given F ε : X → [0, +∞] and F : X → [0, +∞]. F ε Γ -→ F if and only if for every G continuous functional, if x ε minimizes F ε + G and x ε → x then x minimizes F + G .

Our strategy is to replace the singular energy E h with a sequence of smoother elliptic type functionals F ε and prove that F ε Γ -→ E h . Then we prove that the family (F ε ) is equicoercive: any sequence of minima (x j ) is precompact in X. This ensures that the sequence of minimizers xε converge to a minimum. Finally we look for numerical methods to approximate a minimum xε .

Let us present three remarkable examples of Γ-convergence: Modica-Mortola, Ambrosio Tortorelli and a variation of the latter. Consider a container Ω ⊂ R 3 of unitary volume containing two immiscible liquids modeled by a binary function ϕ : Ω → {0, 1} so that Ω |ϕ| dx = V ∈ (0, 1) represents the percentage of one liquid with respect to the container's volume. We associate to the system an energy depending on the surface tension, by supposing that it is directly proportional to the area of the interface J ϕ between the liquids M (ϕ) = cH 2 (J ϕ ).

An alternative way to model this system is to assume that the transition is not given by an infinitesimal separating interface, but is rather a continuous phenomenon occurring in a thin layer of size ε. In view of this Cahn and Hilliard [START_REF] Cahn | Free energy of a nonuniform system. i. interfacial free energy[END_REF] consider a continuous phase function ϕ : Ω → [0, 1] representing the pointwise mixing between the fluids and postulate an energy of the type

Ω ε 2 |∇ϕ| 2 + ϕ 2 (1 -ϕ) 2 dx. ( 13
)
The term ϕ 2 (1ϕ) 2 is called a double well potential and penalizes values far from 0 or 1; inhomogeneity is unfavoured by the gradient term. The link between (12) and (13) was discovered by Modica and Mortola in their papers [START_REF] Modica | Il limite nella Γ-convergenza di una famiglia di funzionali ellittici[END_REF][START_REF] Modica | Un esempio di Γ --convergenza[END_REF]. Their result is more general, as a matter of fact, they prove that a suitable rescaling of the above energy Γ-converges to the perimeter functional in any domain dimension.

Introduction

Let c = 2 1 0 t 2 (1t) 2 dt and M (ϕ) := cH n-1 (J ϕ ), if ϕ = χ A and |A| = V, +∞, otherwise in X.

Then M ε Γ -→ M as ε → 0 in the L 1 topology.

In the above BV(Ω) denotes the space of those functions ϕ such that ϕ ∈ L 1 (Ω) and the distributional gradient Dϕ is a Radon measure. For Bounded Variation functions the distributional gradient can be decomposed into three measures, namely

Dϕ = ∇ϕ + D c ϕ + [ϕ]H n-1 J ϕ
where ∇ϕ is the component of Dϕ absolutely continuous with respect to the Lebesgue measure, D c ϕ is a Cantor measure and [ϕ]H n-1 J ϕ is called the jump component of the measure and is absolutely continuous with respect to the measure Hausdorff measure H n-1 restricted to the discontinuity set J ϕ . In particular if ϕ ∈ BV(Ω) and ϕ = χ A then J ϕ is the essential boundary of A contained in Ω and [ϕ] = 1. For further results on the theory of functions of Bounded Variations we refer to [AFP00] and the technical introduction of Chapter 1, Section 1.2. Theorem 0.7 is correlated with its respective equicoercivity property.

Corollary 0.1. If ε ↓ 0 and ϕ ε minimizes M ε then the sequence (ϕ ε ) si pre-compact with respect to the weak-* topology in BV and any limit point minimizes M .

Another example comes from the approximation of the Mumford-Shah functional for image segmentation. In [START_REF] Mumford | Optimal approximation by piecewise smooth functions and associated variational problems[END_REF] the authors consider a function g, defined on a domain Ω, representing the gray scale values of an image of a group of objects given by a camera, with discontinuities along the edges of the objects. The idea is that the segmented image u should be sufficiently smooth outside an (n -1)-dimensional set containing the discontinuity set K, namely u ∈ W 1,2 (Ω \ K), and the latter should be chosen of minimal H n-1 -size. Therefore they propose to optimize in the variables (u, K) the energy Ω\K |∇u| 2 + α(ug) 2 dx + βH n-1 (K).

The parameters α, β control the weight between the fidelity term |u-g| 2 and the size of the discontinuity set K. It is convenient to recast the problem in its weak formulation letting u ∈ BV(Ω) and replacing the set K with J u obtaining the functional S (u) := Ω |∇u| 2 + α(ug) 2 dx + βH n-1 (J u ).

To give an approximation of the energy S , Ambrosio and Tortorelli have proposed the family of functionals

S ε (u, ϕ) = Ω |∇u| 2 ϕ + β 4 ε|∇ϕ| 2 + (1 -ϕ) 2 ε dx + α

Introduction

In the articles [START_REF] Ambrosio | Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence[END_REF][START_REF] Ambrosio | On the approximation of free discontinuity problems[END_REF] it is proved that S ε Γ -→ S . Let us give a heuristic idea behind this result. Since u is close to g, in the event of a strong discontinuity of g the gradient term |∇u| explodes. Indeed, high values in the gradient |∇u| are controlled by values close to zero in the state function ϕ. On the other hand the term in square brackets strongly penalizes values of ϕ far from 1. The competition of the terms in ϕ results in the fact that 1ϕ represents a smoothed version of the function 1χ Ju . Finally in the limit ε ↓ 0 the Modica-Mortola term converges to the H n-1 size of the set {ϕ = 1} which contains the jump set of u. Functionals modeled on the ones from Ambrosio and Tortorelli and the latter functional itself are frequently known under the name of phase-field approximations. This is not only because of the strict relation with the Modica-Mortola functional but even because we may interpret the function ϕ as a state function, which acquires value 0 on the jump set of u, i.e. on the set of strong discontinuity of the function, and value 1 where u is sufficiently smooth. The two behaviors of u are then interpreted as two possible states and ϕ models the pointwise state function for the system. This observation has been taken into consideration in the work on fracture theory from Iurlano et al. [START_REF] Conti | Phase field approximation of cohesive fracture models[END_REF][START_REF] Iurlano | Fracture and plastic models as Γ-limits of damage models under different regimes[END_REF]. There ϕ models the damage state of a material and u is replaced with a displacement function.

To conclude the section we present a variation of the Ambrosio Tortorelli functional proposed by Bonnivard, Lemenant and Santambrogio [START_REF] Lemenant | A Modica-Mortola approximation for the Steiner problem[END_REF][START_REF] Bonnivard | Approximation of length minimization problems among compact connected sets[END_REF] to recover in the limit the functional associated to the Steiner minimal tree problem for some points {x 0 , . . . , x N } ⊂ Ω ⊂ R 2 . Given a continuous function ϕ : Ω → [0, 1] the authors introduce a geodesic distance weighted on ϕ, namely

d ϕ (x, y) = inf γ ϕ dH 1 : γ ∈ C([0, 1], Ω), γ(0) = x, γ(1) = y .
The distance d ϕ (x, y) vanishes if and only if the two points x, y are joined by a path on which ϕ is equal to 0. Now consider the functional

Ω ε|∇ϕ| 2 + (1 -ϕ) 2 4ε dx + 1 c ε N i=1 d ϕ (x 0 , x i )
where c ε → 0 as ε → 0. First observe that if

N i=1 d ϕ (x 0 , x i ) = 0 (14)
then the set {ϕ = 0} should include a path-connected subset containing {x 0 , . . . , x N }.

The heuristic argument for the Γ-convergence result follows the ideas presented in the case of the Ambrosio-Tortorelli functional. The exact result in [START_REF] Bonnivard | Approximation of length minimization problems among compact connected sets[END_REF] is

Theorem 0.8 (Bonnivard-Lemenant-Santambrogio). Let Ω ⊂ R 2 be an open set, {x 0 , . . . , x N } ⊂ Ω and µ = 1 N N i=1 δ x i . Consider the functional B ε (ϕ) = Ω ε|∇ϕ| 2 + (1 -ϕ) 2 4ε dx + Introduction and a sequence ϕ ε such that B ε (ϕ ε ) -inf ϕ B ε (ϕ) --→ ε→0 0.
Then the sequence of functions d ϕε converges uniformly (up to a subsequence) to a function d such that the set K := {d = 0} minimizes H 1 among all compact, connected sets containing the points {x 0 , . . . , x n }.

A first approach to the problem of approximating the energy E h in the case h = | • | α was proposed by Santambrogio and Oudet in [START_REF] Oudet | A Modica-Mortola approximation for branched transport and applications[END_REF]. They introduce a functional of the type

Ω ε α+1 |∇σ| 2 + ε α-1 |σ| β with σ ∈ W 1,2 (Ω, R 2 ) and ∇ • σ = (µ + -µ -) * ρ ε
with β = (4α -2)/(α + 1) and ρ ε an approximation to the identity. Actually the complete Γlim sup inequality for the latter result has been provided by Monteil [START_REF] Monteil | Approximations elliptiques d'nergies singulires sous contrainte de divergence[END_REF][START_REF] Monteil | Uniform estimates for a Modica-Mortola type approximation of branched transportation[END_REF].

Structure of the thesis

In the First Chapter we study a variation of the functional proposed by Lemenant and Santambrogio. Motivated by the observation that

d ϕ (x, y) = min Ω ϕ|σ| dx : σ ∈ M(Ω, R n ) and div σ = δ x -δ y
we replace the term depending on the geodesic distance in (15) with a term depending on the product ϕ|σ|. The proposed functional is defined on couples (σ, ϕ) is

Ω ϕ|σ| 2 2ε dx + Ω ε 2 |∇ϕ| 2 + (1 -ϕ) 2 2ε dx,
where σ is a vector-valued function complemented with the constraint

div σ = (µ + -µ -) * ρ ε . (16)
In the above ρ ε is a given approximation of the identity and the phase functions ϕ ∈ L 1 (Ω) are bounded from below by the quantity β ε, where β ≥ 0 a given parameter. First, we show that this functional Γ-converges to the energy E h , for the choice

h(m) := 1 + βm, if m = 0, 0, otherwise. ( 17
)
The proof of the Γ-convergence result is obtained for open convex subsets of R 2 . The advantage of choosing a quadratic penalization in σ is that the augmented Lagrangian problem associated to the functional may be explicitly solved in the dual variable. Therefore it is possible to devise an alternate minimization algorithm composed of two smooth elliptic functionals solvable via finite elements methods. The algorithm is

Introduction

proposed and studied at the end of the chapter. We further present and study other algorithms which take advantage of a concept of 'shape derivative' to improve the quality of the approximation.

The generalization to Ω ⊂ R n is the matter of the Second Chapter. To obtain the result in higher dimension the Modica-Mortola component of the functional needs to be rescaled. As observed in [START_REF] Ghiraldin | Variational approximation of a functional of Mumford-Shah type in codimension higher than one[END_REF] this leads to the introduction of some non linearities in the functional as follows

Ω ϕ|σ| 2 ε dx + Ω ε p-n+1 |∇ϕ| p + (1 -ϕ) 2 ε n-1 dx, (18)
for some p > n -1. Again σ is complemented with the divergence constraint (16) for a suitable choice of ρ ε and we require a lower bound for the the phase field functions, namely ϕ ≥ βε n . We prove the Γ-convergence of the above functional to E h n-1 β where the cost function h n-1 β is the limit in ε of an optimization problem depending on the co-dimension n -1. Namely for a ball B r ⊂ R n-1 we let

h n-1 ε,β (m) = min        Br ϕ|θ| 2 ε + ε p-n+1 |∇ϕ| p + (1 -ϕ) 2 ε n-1 dx, ϕ ∈ W 1,p (B r ), ϕ = 1 on ∂B r and Br θ dx = m.
The latter optimization problem corresponds to the 0-dimensional version of (18). We introduce and study h d ε,β (obtained replacing n -1 with d in the latter formula) in the appendix. Some similar phase transition problems with mass constraint which leads to measures concentrated on atoms have been studied by Bouchitté, Dubs and Seppecher in [START_REF] Bouchitté | Transitions de phases avec un potentiel dégénéré à l'infini, application à l'équilibre de petites gouttes[END_REF] in the context of droplets equilibrium. In particular we show that h d β is independent of r and that it is a transport cost function satisfying the conditions (3). We prove as well that there exists a constant c > 0 such that

1 c ≤ h d β (m) 1 + √ βm ≤ c for m > 0.
Remark that the Modica-Mortola component of the functional studied in the second chapter depends on n -1, the co-dimension of the problem in the case of rectifiable measures. In Chapter Three we investigate a different rescaling to approach minima to (11) defined for k-currents, namely

Ω ϕ|σ| 2 ε dx + Ω ε p-n+k |∇ϕ| p + (1 -ϕ) 2 ε n-k dx. ( 19
)
In this context, σ is no longer a vector measure, to take into account the boundary the constraint needs to be suitably modified. Let σ 0 be a given polyhedral k-current, for ρ ε a standard approximation of the identity we let σ be a a smooth k-current such that ∂σ = ∂σ 0 * ρ ε .

(In equation ( 19) the current is identified with its density measure.) In the chapter we introduce formally the energy and show that it Γ-converges to the energy E h defined in (11) for the transport cost function h = h n-k β studied in the appendix. In the Fourth and Fifth Chapters we restrict again our attention to sets Ω ⊂ R 2 and develop two functionals for the approximation of any concave and continuous transport cost function h. Note that we say that a transport cost function is concave if it is an even function whose restriction to [0, +∞) is concave. The first result regards transport cost functions h of the form

h(m) = min{α i |m| + β i : 0 ≤ i ≤ N }. for α 0 > α 1 > . . . > α N ≥ 0 and 0 ≤ β 0 < β 1 < . . . < β N .
Our approach takes advantage of the result in the First Chapter in which we recovered in the Γ-limit affine cost functions of the form 1 + β|m|. In the case N = 1 and β 0 > 0 the proposed phase-field energy takes the form

Ω min ϕ 2 0 + α 2 0 ε 2 β 0 ; ϕ 2 1 + α 2 1 ε 2 β 1 |σ| 2 2ε dx + β 0 T ε (ϕ 0 ) + β 1 T ε (ϕ 1 )
where T ε is an energy of the Modica-Mortola type defined as

T ε (ϕ) = 1 2 Ω ε|∇ϕ(x)| 2 + (ϕ(x) -1) 2 ε dx.
Let us highlight the presence of two phase-fields which interact in the constraint component of the functional. Ideally each 1-ϕ i is a smooth indicator function of some subset of the support of the limit rectifiable measure σ. In particular ϕ i = 0 if the choice of the i-th component in the definition of h is optimal with respect to the intensity of the flux of σ. The entire Fourth Chapter is devoted to establish the proof of the Γ-convergence result and the study of numerical methods developed in collaboration with Carolin Rossmanith and Benedikt Wirth from Munster University.

In the final chapter of the thesis we study functionals of the form

F ε (σ, ϕ) := Ω f (ϕ)|σ| + 1 2 ε|∇ϕ| 2 + ϕ 2 ε dx
The two main differences with respect to the previous models are the linear penalization in |σ| and the presence of the term ϕ 2 instead of (1-ϕ) 2 . Analogous models with a linear penalization of the |σ| component have been studied recently in the case of fracture theory and the generalized Mumford-Shah functional [START_REF] Alicandro | Free-discontinuity problems via functionals involving the L 1 -norm of the gradient and their approximations[END_REF][START_REF] Maso | Fracture models for elastoplastic materials as limits of gradient damage models coupled with plasticity: the antiplane case[END_REF]. Our main Introduction contribution is to find an explicit form of the weight function f to obtain in the limit the energy E h . For a continuous and concave transport cost function h, we define f as

f (t) = (-h * ) -1 (t 2 ).
The function h * is the (concave) Legendre transform of h. In this model ϕ takes value 0 and not 1 outside the support of the limit measure σ. By virtue of this general result we address the problem of the numerical approximation of the functional F ε .

The linear penalization in σ may be seen as a drawback with respect to the methods previously studied which where deeply based on the quadratic cost |σ| 2 . In view of this difference we started investigating new numerical methods based on the Beckman model [START_REF] Beckmann | A continuous model of transportation[END_REF] for transportation. The same result may be obtained with different choice of the well potential. Namely, given a potential W which is an even function, increasing on [0, +∞) and vanishing in 0 we introduce the transition energy

c W (t) := |t| 0 2 W (s) ds.
Then choosing f (t) = (-h * ) -1 • c W (t) the same Γ-convergence result may be obtained with a family of functionals defined as

F ε (σ, ϕ) := Ω f (ϕ)|σ| + 1 2 ε|∇ϕ| 2 + W (ϕ) ε dx.
In force of this degree of freedom in the choice of the potential W we start analyzing which would be the best choice. These and other questions are the subject of the concluding section which investigates possible developments of the proposed methods.

Chapter 1

Affine cost function

Introduction

In this chapter we devise an approximation for the minimization problem defined in the introduction in equation ( 6). In particular we consider the energy E h choosing as cost function

h(m) := 1 + β|m|, if m = 0 0, otherwise,
where β > 0 is a fixed positive parameter. Furthermore we will consider only atomic probability measures µ + , µ -∈ P(Ω) to define the constraint. The results contained in this chapter have been published in the paper [START_REF] Chambolle | A phase-field approximation of the steiner problem in dimension two[END_REF]. Let us introduce the precise framework of our approximation. Let ρ ∈ C ∞ c (R 2 , R + ) be a classical radial mollifier with supp ρ ⊂ B 1 (0) and ρ = 1. For ε ∈ (0, 1], we set ρ ε (x) = ε -2 ρ(ε -1 x) and we define the space V ε (Ω) of square integrable vector fields with weak divergence satisfying the constraint

∇ • σ ε = (µ + -µ -) * ρ ε in D (R 2 ). (1.1)
For an η = η(ε) > 0, we denote

W ε (Ω) = ϕ ∈ W 1,2 (Ω) : η ≤ ϕ ≤ 1 in Ω, ϕ ≡ 1 on ∂Ω .
We denote with X ε (Ω) = V ε (Ω) × W ε (Ω) and define the energy

F ε : M(Ω, R 2 ) × L 1 (Ω) → [0, +∞] as F ε (σ, ϕ) :=      Ω 1 2ε ϕ 2 |σ| 2 dx + Ω ε 2 |∇ϕ| 2 + (1 -ϕ) 2 2ε dx, if (σ, ϕ) ∈ X ε (Ω), +∞ , otherwise.
(1.2)

From now on, we assume that

η ε ε↓0 -→ β. (1.3) We denote M S (Ω) the set of R 2 -valued measures σ ∈ M(R 2 , R 2 ) with support in Ω such that the constraint div σ = µ + -µ -in D (R 2 ) (1.4)
Chapter 1. Affine cost function holds. We define the limit energy

E β : M(Ω, R 2 ) × L 1 (Ω) → [0, +∞] as E β (σ, ϕ) =    Σ (1 + β m) dH 1 if ϕ ≡ 1, σ ∈ M S (Ω) and σ = (m, τ, Σ), + ∞ otherwise.
(1.5)

We prove the Γ-convergence of the sequence (F ε ) to the energy E β as ε ↓ 0. More precisely the convergence holds in M(Ω, R 2 ) × L 1 (Ω) where M(Ω, R 2 ) is endowed with the weak- * topology and L 1 (Ω) is endowed with its classical strong topology. We begin by proving the equicoercivity of the sequence (F ε ). In this statement and throughout the chapter, we make a small abuse of language by denoting (a ε ) ε∈(0,1] and calling sequence a family {a ε } labeled by a continuous parameter ε ∈ (0, 1]. In the same spirit, we call subsequence of (a ε ), any sequence (a ε j ) with ε j → 0 as j → +∞. We establish the following lower bound.

Theorem 1.1 (Γ -lim inf). For any sequence (σ ε , ϕ ε ) ⊂ M(Ω, R 2 ) × L 1 (Ω) such that σ ε * σ and ϕ ε → ϕ in the L 1 (Ω) topology, with (σ, ϕ) ∈ M(Ω, R 2 ) × L 1 (Ω), lim inf k→+∞ F ε (σ ε , ϕ ε ) ≥ E β (σ, ϕ).
To complete the Γ-convergence analysis, we establish the matching Γ-limsup inequality.

Theorem 1.2 (Γ-lim sup). For any (σ, ϕ) ⊂ M(Ω, R 2 )×L 1 (Ω) there exists a sequence (σ ε , ϕ ε ) such that σ ε * σ and ϕ ε → ϕ in the L 1 (Ω) topology and

lim sup k→+∞ F ε (σ ε , ϕ ε ) ≤ E β (σ, ϕ). Theorem 1.3 (Equicoercivity). Assume β > 0. For any sequence (σ ε , ϕ ε ) ε∈(0,1] ⊂ M(Ω, R 2 ) × L 1 (Ω) with uniformly bounded energies, i.e. sup ε F ε (σ ε , ϕ ε ) < +∞,
there exist a subsequence ε j ↓ 0 and a measure σ ∈ M S (Ω, R 2 ) such that σ ε j → σ with respect to the weak- * convergence of measures and ϕ ε j → 1 in L 1 (Ω). Moreover, σ is a rectifiable measure (i.e., it is of the form σ = (m, τ, Σ)).

Structure of the chapter:

In Section 1.2 we introduce and recall some notation and several tools and notions on SBV functions and introduce an operator acting on vector field measures. In Section 1.3 we study the behavior of the functional F ε on vector fields of the form ∇u (dropping the divergence constraint). In Section 1.4 we prove the equicoercivity result, Theorem 1.3 and we establish the lower bound stated in Theorem 1.1. In Section 1.5 we prove the upper bound of Theorem 1.2. Finally, in the last section, we present and discuss various numerical simulations. BV(Ω) is the space of functions u ∈ L 1 (Ω) having as distributional derivative Du a measure with finite total variation. Following the classical notation as in [AFP00, ABM14] and [START_REF] Braides | Approximation of free-discontinuity problems[END_REF] for u ∈ BV (Ω) we have

Du = ∇u dx + (u + -u -)ν u H d-1 J u + D c u,
where J u is the set of "approximate jump points" x where y → u(x + ρy) converge as ρ → 0 to u + χ {y•νu≥0} + u -χ {y•νu<0} for some (u -, u + , ν u ) and D c u is the Cantor "part". Let us introduce the space of special functions of bounded variation and a variant:

SBV (Ω) := {u ∈ BV (Ω) : D c u = 0}, GSBV (Ω) := {u ∈ L 1 (Ω) : max(-T, min(u, T )) ∈ SBV (Ω) ∀ T > 0}.
Eventually, in Section 1.3, the following space of piecewise constant functions will be useful.

P C(Ω) = {u ∈ GSBV (Ω) : ∇u = 0}.

(1.6)

To conclude this section we recall the slicing method for functions of bounded variation. Let τ ∈ S d-1 and let

Π τ := {y ∈ R d : y, τ = 0}.
If y ∈ Π τ and E ⊂ R d , we define the one dimensional slice

E τ,y := {t ∈ R : y + tτ ∈ E}.
For u : Ω → R, we define u τ,y : Ω τ,y → R as

u τ,y (t) := u(y + tτ ), t ∈ Ω τ,y .
Functions in GSBV (Ω) can be characterized by one-dimensional slices (see [START_REF] Braides | Approximation of free-discontinuity problems[END_REF]Thm. 4.1])

Chapter 1. Affine cost function Theorem 1.4. Let u ∈ GSBV (Ω). Then for all τ ∈ S d-1 we have

u τ,y ∈ GSBV (Ω τ,y ) for H d-1 -a.e. y ∈ Π τ .
Moreover for such y, we have

u τ,y (t) = ∇u(y + tτ ), τ for a.e. t ∈ Ω τ,y , J uτ,y = {t ∈ R : y + tτ ∈ J u }, and u τ,y (t ± ) = u ± (y + tτ ) or u τ,y (t ± ) = u ∓ (y + tτ )
according to whether ν u , τ > 0 or ν u , τ < 0. Finally, for every Borel function

g : Ω → R, Πτ t∈Ju τ,y g τ,y (t) dH d-1 (y) = Ju g| ν u , τ | dH d-1 .
(1.7)

Conversely if u ∈ L 1 (Ω)
and if for all τ ∈ {e 1 , . . . , e d } and almost every y ∈ Π τ we have u τ,y ∈ SBV (Ω τ,y ) and

Πτ |Du τ,y |(Ω τ,y ) dH d-1 (y) < +∞ then u ∈ SBV (Ω).
Let us introduce the linear operator ⊥ that associates to each vector v = (v 1 , v 2 ) ∈ R 2 the vector v ⊥ = (-v 2 , v 1 ) obtained via a 90 • counterclockwise rotation of v. Notice that the ⊥ operator maps divergence-free R 2 -valued measures onto curl free R 2 -valued measures. Let O ⊂ R 2 be a simply connected and bounded open set. It is possible to generalize Stokes Theorem to divergence free measures. If µ is a smooth divergence free vector field on O we have µ = ∇u ⊥ for some smooth function with zero mean value. Then by Poincaré inequality |u| L 1 ≤ C|µ| L 1 . The result for µ general divergence free finite vector measure follows by regularization. On the other hand for u ∈ P C(Ω), σ := Du ⊥ is divergence free and,

σ = (u + -u -)ν ⊥ u H 1 = U (J u , [u], ν ⊥ u ).
(1.8)

Local Result

In this section we introduce a localization of the family of functionals (F ε ) (see (1.2)). We establish a lower bound and a compactness property for these local energies. In this section we assume Ω ⊂ R d .

Localization. Let O ∈ A S (Ω) be a simply connected relatively open subset of Ω. For

u ε ∈ W 1,2 (O) and ϕ ε ∈ W 1,2 (O), we define G ε (u ε , ϕ ε ; O) := F ε (∇u ε , ϕ ε ; O) 1.3. Local Result
i.e. as the evaluation of the functional F ε on vector fields of the form ∇u with no requirement on the divergence. Notice that for ε < d(O, S), we have

∇ • σ ε ≡ 0 in O for any σ ε ∈ V ε (Ω). By Stokes theorem we have Du ⊥ ε = σ ε for some u ε ∈ W 1,2 (O) and we have F ε (σ ε , ϕ ε ; O) = G ε (u ε , ϕ ε ; O).
The rest of the section is devoted to the proof of Theorem 1.5. Let (u ε ) ε∈(0,1] ⊂ W 1,2 (O) be a family of functions with zero mean value and let

(ϕ ε ) ⊂ W 1,2 (O) such that ϕ ε ∈ W 1,2 (O, [η(ε), 1]). Assume that c 0 := sup ε G ε (u ε , ϕ ε ; O) is finite.
Then there exist a subsequence ε j and a function u

∈ BV (Ω) such that a) ϕ ε j → 1 in L 2 (O), b) u ε j → u with respect to the weak- * convergence in BV , c) u ∈ P C(O).
Furthermore for any piecewise function u ∈ P C(O) and any sequence (u ε , ϕ ε ) such that u ε * u and ϕ ε → 1, we have the following lower bound of the energy:

lim inf ε→0 G ε (u ε , ϕ ε ; O) ≥ Ju∩O (1 + β|[u]|) dH d-1 .
The proof is achieved in several steps and mostly follows ideas from [START_REF] Iurlano | Fracture and plastic models as Γ-limits of damage models under different regimes[END_REF] (see also [START_REF] Conti | Phase field approximation of cohesive fracture models[END_REF]). In the first step we obtain (a) and (b). In step 2 we prove (c) and the lower bound for one dimensional slices of G ε . Finally in step 3 we prove (c) and the lower bound in dimension d. The construction of a recovery sequence that would complete the Γ-limit analysis is postponed to the global model in Section 1.5.

Proof.

Step 1: Item (a) is a straightforward consequence of the definition of the functional. Indeed, we have

O (1 -ϕ ε ) 2 dx ≤ 2 ε G ε (u ε , ϕ ε ) ≤ 2, c 0 ε ε↓0 -→ 0.
For (b), since (u ε ) has zero mean value, we only need to show that sup ε∈(0,1] |Du ε |(O) < +∞. Using Cauchy-Schwarz inequality we get

[|Du ε |(O)] 2 = O |∇u ε | 2 ≤ 2 ε O 1 ϕ 2 ε 1 ε O ϕ 2 ε |∇u ε | 2 .
(1.9)

By assumption, the second therm in the right hand side of (1.9) is bounded by 2c 0 . In order to estimate the first term we split O in the two sets {ϕ ε < 1/2} and {ϕ ε ≥ 1/2}. We have,

2 ε O 1 ϕ 2 ε = 2 ε {ϕε<1/2} 1 ϕ 2 ε + {ϕε≥1/2} 1 ϕ 2 ε . Chapter 1. Affine cost function Since η ≤ ϕ ε ≤ 1/2 on {ϕ ε ≤ 1/2} it holds ϕ 2 ε (1 -ϕ ε ) 2 ≥ η 2 (1 -1/2) 2 therefore {ϕε<1/2} 1 ϕ 2 ε ≤ 2ε η 2 (1 -1/2) 2 {ϕε<1/2} (1 -ϕ ε ) 2 2ε ≤ 8ε η 2 c 0 , {ϕε≥1/2} 1 ϕ 2 ε ≤ {ϕε≥1/2} 1 (1/2) 2 = 4|{ϕ ε ≥ 1/2}|.
Eventually, as |{ϕ ε ≥ 1/2}| ≤ |O|, combining these estimates with (1.9) we obtain

[|Du ε |(O)] 2 ≤ ε 2 η 2 16c 2 0 + 8ε|O|c 0 ε↓0 -→ 16c 2 0 β 2 < ∞. (1.10)
This establishes (b).

Step 2: In this step we suppose O to be an interval of R, so that u ε , ϕ ε are onedimensional. We first prove that u is piecewise constant. The idea is that in view of the constraint component of the energy, variations of u ε are balanced by low values of ϕ ε . On the other hand the Modica-Mortola component of the energy implies that ϕ ε 1 in most of the domain and that transitions from ϕ ε 1 to ϕ ε 0 have a constant positive cost (and therefore can occur only finitely many times).

Step 2.1: (Proof of u ∈ P C(O).) Let us define

B ε := x ∈ O : ϕ ε (x) < 3 4 ⊃ A ε := x ∈ O : ϕ ε (x) < 1 2 , (1.11)
G ε (u ε , ϕ ε ; I) ≥ I ε|ϕ ε | 2 2 + (1 -ϕ ε ) 2 2ε dx ≥ (a,b) |ϕ ε |(1 -ϕ ε ) dx ≥ 3/4 1/2 (1 -t) dt = 3 2 5 .
Since all the elements of C ε are disjoint and G ε (u ε , ϕ ε , •) is additive, we deduce from the energy bound that #C ε ≤ 2 5 c 0 /3, where we denote #C ε the cardinality of C ε . Next, up to extracting a subsequence we assume that #C ε = N is fixed. The elements of C ε are of the form

I ε i = (m ε i -w ε i , m ε i + w ε i ) for i = 1, • • • , N , with m ε i < m ε i+1 . Since ϕ ε → 1 in L 1 (O)
we have

I ε i ∈Cε |I ε i | = N i=1 2w ε i → 0.
(1.13)

Local Result

Up to extracting a subsequence, we can assume that each sequence (m ε i ) converges in O. We call m 1 ≤ m 2 ≤ • • • ≤ m N their respective limits. We now prove that

|Du|(O \ {m i } N i=1 ) = 0, (1.14) thus supp(|Du|) ⊂ {m 1 , • • • , m N }.
The latter ensures that u has no Cantor component since Du is supported on a finite number of points and that is a.e. constant outside {m 1 , • • • , m N } so that u ∈ P C(O), (1.6). To this aim, we fix x ∈ O \ {m i } N i=0 and establish the existence of a neighborhood B δ (x) of x for which |Du|(B δ (x)) = 0. Let 0 < δ ≤ min i |x-m i |/2. Equation (1.13) ensures that for ε small enough B δ (x)∩C ε = ∅. Notice that from the definitions in (1.11) and (1.12) we have that ϕ ε ≥ 1/2 outside the union of the sets in C ε . Hence, using Cauchy-Schwarz inequality, we have for ε small enough,

B δ (x) |u ε | dx 2 ≤ 2δ B δ (x) |u ε | 2 dx ≤ (2δ)(2ε)4 1 2ε B δ (x) ϕ 2 ε |u ε | 2 dx ≤ 16c 0 εδ ε↓0 -→ 0.
By lower semicontinuity of the total variation on open sets we conclude that |Du|(B δ (x)) = 0, which proves the claim (1.14).

Step 2.2: (Proof of the lower bound for G ε .) Without loss of generality we can assume N = 1, thus J u is composed of a single point, otherwise the argument we propose can be applied on each m i separately. Up to a translation m 1 = 0 and we denote D := u(0 + ) = -u(0 -) > 0. For any 0 < d < D there exist six points

y 1 < x 1 ε ≤ x1 ε < x2 ε ≤ x 2 ε < y 2 such that lim ε→0 ϕ ε (y 1 ) = lim ε→0 ϕ ε (y 2 ) = 1, lim ε→0 ϕ ε (x 1 ε ) = lim ε→0 ϕ ε (x 2 ε ) = 0, (1.15) u ε (x 1 ε ) = -D + d, u ε (x 2 ε ) = D -d.
Since ϕ ε → ϕ and u ε → u in L 1 up to a subsequence they converge point-wise almost everywhere and this implies the first and third fact. Let inf (y 1 ,y 2 ) ϕ ε = c ε , then Jensen inequality implies

c 0 ≥ y 2 y 1 ϕ 2 ε |u ε | 2 2ε dx ≥ c 2 ε 2ε(y 2 -y 1 ) y 2 y 1 |u ε | dx 2 .
Then c ε must vanish with ε implying statement (1.15). Using the Modica-Mortola trick in the intervals (y 1 , x 1 ε ) and (x 2 ε , y 2 ) as above, we compute:

lim inf ε↓0 G ε (u ε , ϕ ε ; (y 1 , x 1 ε ) ∪ (x 2 ε , y 2 )) ≥ ≥ lim inf ε↓0 x 1 ε y 1 (1 -ϕ ε )|ϕ ε | dx + y 2 x 2 ε (1 -ϕ ε )|ϕ ε | dx ≥ 1.
(1.16)

Chapter 1. Affine cost function

For the estimate on the interval I ε = (x 1 ε , x2 ε ) let us introduce:

G ε := w ∈ W 1,2 (I ε ) : w(x 1 ε ) = -D + d, w(x 2 ε ) = D -d , Z ε := z ∈ W 1,2 (I ε ) : η ≤ z ≤ 1 a.e. on I ε , H ε (w, z) := Iε 1 2ε z 2 |w | 2 + (1 -z) 2 2ε dx, h ε (z) = inf w∈Gε H ε (w, z) for z ∈ Z ε .
Note that for w ∈ G ε and z ∈ Z ε , we can apply an inequality similar to (1.9). In particular, for z replacing ϕ ε and w taking the place of Du ε we get

Iε |w | dx 2 ≤ Iε z 2 |w | 2 Iε 1 z 2 .
Reversing the latter and taking into account the conditions on w obtains

Iε z 2 |w | 2 ≥ Iε |w | dx 2 Iε 1 z 2 -1 ≥ 4(D -d) 2 Iε 1 z 2 -1
.

From this we deduce the lower bound

h ε (z) ≥ 4(D -d) 2 2ε Iε 1 z 2 dx -1 + Iε (1 -z) 2 2ε dx.
(1.17)

Let us remark that optimizing H ε (w, z) with respect to w ∈ G ε we see that this inequality is actually an equality. Consider for 0 < λ < 1 the inequalities:

{x∈Iε:ϕε≥λ} 1 ϕ 2 ε ≤ |I ε | λ 2 , {x∈Iε:ϕε<λ} 1 ϕ 2 ε ≤ 1 (1 -λ) 2 2ε η 2 Iε (1 -ϕ ε ) 2 2ε dx .
Applying both of them in (1.17) we obtain

G ε (u ε , ϕ ε , I ε ) ≥ h ε (ϕ ε ) ≥ 2(D -d) 2 ε|Iε| λ 2 + 1 (1-λ) 2 2ε 2 η 2 Iε (1-ϕε) 2 2ε dx + Iε (1 -ϕ ε ) 2 2ε dx ≥ 2(1 -λ) η ε (D -d) -(1 -λ) 2 η 2 2ε |I ε | λ 2 (1.18)
where the latter inequality is obtained by minimizing the function:

t → 2(D -d) 2 ε|Iε| λ 2 + 1 (1-λ) 2 2ε 2 η 2 t + t.
Therefore we can pass to the limit in (1.18) and obtain:

lim inf ε↓0 G ε (u ε , ϕ ε , I ε ) ≥ (1 -λ)β 2(D -d).

Local Result

Sending λ and d to 0 and recalling the estimate in (1.16) we get

lim inf ε↓0 G ε (u ε , ϕ ε , (y 1 , y 2 )) ≥ 1 + β 2D = 1 + β|u(0 + ) -u(0 -)|. (1.19)
Step 3: Indeed by Fatou's Lemma for any τ ∈ S d-1 and H d-1 almost every y ∈ Ω τ it holds

lim inf ε↓0 G ε (u ε , ϕ ε ; O) ≥ Πτ lim inf ε↓0    O τ y 1 2ε (ϕ 2 ε ) τ y |(u ε ) τ y | 2 + ε 2 |(ϕ ε ) τ y | 2 + (1 -(ϕ ε ) τ y ) 2 2ε dt    dH d-1 (y).
Then by the results in Step 2.1 and 2.2, in particular inequality (1.19), it holds

lim inf ε↓0 G ε (u ε , ϕ ε ; O) ≥ Πτ m i ∈(Ju) τ y 1 + β|u τ y (m + i ) -u τ y (m - i )| dH d-1 (y).
Therefore by Theorem 1.4 we have u ∈ SBV (O). Moreover, since (u ) τ y = 0 on each slice, we have u ∈ P C(O). Applying identity (1.7) we get

lim inf ε→0 G ε (u ε , ϕ ε ; O) ≥ Ju∩O |ν u • τ | [1 + β|[u]|] dH d-1 .
(1.20)

In order to conclude, we use the following localization method stated by Braides in [Bra98, Prop. 1.16].

Lemma 1.1. Let µ : A(X) → [0, +∞) be an open-set function superadditive on open sets with disjoint compact closures and let λ be a positive measure on X. For any i ∈ N let ψ i be a Borel function on X such that µ(A) ≥ A ψ i dλ for all A ∈ A(X). Then

µ(A) ≥ A ψ dλ
where ψ := sup i ψ i .

For any u ∈ P C(O) let us introduce the increasing set function µ defined on A(O) by

µ(A) := inf (ϕε,uε)→(1,u) lim inf ε→0 G ε (u ε , ϕ ε ; A) , for any A ∈ A(O).
Observe that for any two open sets A and B with disjoint compact closure and for any (u ε , ϕ ε ) such that u ε * u and ϕ ε → 1 on A ∪ B, the restriction of u ε to A (resp. B)

weak- * converges in A (resp. B) to the restriction of u on A (resp. B) and it follows

µ(A + B) ≥ µ(A) + µ(B).
This proves that µ is superaddittive on open sets with disjoint compact closures. Let λ be a Radon measure defined as

λ := [1 + β|u(x + ) -u(x -)|]H d-1 J u .
Chapter 1. Affine cost function Fix a sequence (τ i ) i∈N dense in S d-1 . By (1.20) we have

µ(O) ≥ O ψ i dλ, i ∈ N,
where

ψ i (x) := | ν u (x), τ i |, if x ∈ J u , 0, if x ∈ O \ J u .
Hence by Lemma 1.1 we finally obtain

lim inf ε→0 G ε (u ε , ϕ ε ; O) ≥ O sup i ψ i (x) dµ = Ju∩O [1 + β|[u]|] dH d-1 .

Equicoercivity and Γ-liminf

From now till the end of the chapter we assume that Ω ⊂ R 2 . Let us first produce the following construction.

Lemma 1.2. Given two probability measures µ + and µ -supported on a finite set of points S = {x 0 , . . . , x N }, there exists a vector measure γ = U (m γ , τ γ , Σ γ ) and a finite partition

(Ω i ) ⊂ A(Ω) of Ω such that a) ∇ • γ = -µ + + µ -, b) each Ω i is a polyhedron, c) Σ γ ⊂ i ∂Ω i , d
) Ω i is of finite perimeter for each i and

Ω i ∩ Ω j = ∅ for i = j, e) |Ω \ ∪ i Ω i | = 0.
Moreover if M is a 1 dimensional countably rectifiable set, we can choose γ and

(Ω i ) such that H 1 (M ∩ i ∂Ω i ) = 0.
Proof. Let us fix a point p ∈ Ω \ S and assume

µ + -µ -= N i=0 a i δ x i .
Consider the map x i -p |x i -p| t + p : [0, 1] → Ω, then the measure

γ i = p -x i |p -x i | • +x i # [0, 1] is supported on the segment [p, x i ] =: Σ γ and is such that ∇ • γ i = δ x i -δ p for i ∈ {0, • • • , N }. We define γ = N i=1 a i γ i .
1.4. Equicoercivity and Γ-liminf By construction (a) holds true. Moreover, up to a small shift of p we may assume that [p, x i ] ∩ [p, x j ] = {p} for i = j.

Next, let D j be the straight line supporting [p, x j ]. We define the sets (Ω i ) as the connected components of Ω \ (D 0 ∪ • • • ∪ D N ). We see that (c, d, e) hold true.

For the last statement, we observe that by the coarea formula, we have H 1 (Σ∩ i ∂Ω i ) = 0 for a.e. choice of p. We now prove the compactness property (Theorem 1.3). Let us consider a sequence (σ ε , ϕ ε ) ∈ M(Ω, R 2 ) uniformly bounded in energy by c 0 < +∞,

0 ≤ F ε (σ ε , ϕ ε ) ≤ c 0 for ε ∈ (0, 1]. (1.21)
Proof of Theorem 1.3. First observe that by definition (1.2) and equation (1.21), we have σ ε ∈ V ε (Ω) and ϕ ε ∈ W ε (Ω). Next, using the arguments of Step 1 of the proof of Theorem 1.5, with |σ ε | instead of |∇u ε |, inequality (1.10) reads

|σ ε |(Ω) ≤ 16 ε 2 η 2 c 2 0 + 8ε|Ω|c 0 ε↓0 -→ 4c 0 β < ∞. (1.22)
Thus the total variation of (σ ε ) ε is uniformly bounded as long as β > 0 and there exists a σ ∈ M S (Ω) such that up to a subsequence σ ε * σ in M(Ω).

Now, considering the last term in the energy (1.2) we have

Ω (1 -ϕ ε ) 2 dx ≤ 2ε F ε (σ ε , ϕ ε ) ≤ 2ε c 0 → 0. Hence, ϕ ε → 1 in L 2 (Ω).
Let us now study the structure of the limit measure σ. Let us recall that Ω is a bounded convex relatively open set such that Ω ⊂ Ω and let us extend σ ε by 0 and Chapter 1. Affine cost function

ϕ ε by 1 in Ω \ Ω. Obviously we have F ε (σ ε , ϕ ε ; Ω) = F ε (σ ε , ϕ ε ; Ω),
σ O = Du ⊥ O = -[u]ν ⊥ Ju H 1 (J u ∩ O).
Since we can cover Ω \ S by countable many sets O ∈ A S ( Ω), this shows that σ decomposes as

σ = (m σ , τ σ , Σ σ ) + ω,
where ω is a measure absolutely continuous with respect to H 0 S. By Lemma 1.2 there exists a rectifiable measure γ = U (m γ , τ γ , Σ γ ) such that ∇ • (σ + γ) = 0 and H 1 (Σ γ ∩ Σ σ ) = 0. Then there exists a u ∈ BV (Ω) such that Du = σ ⊥ + γ ⊥ . Since u ∈ BV (Ω) and S is composed by a finite number of points, we deduce |Du|(S) = 0 which implies |ω|(S) = 0. Hence σ writes in the form (m σ , τ σ , Σ σ ).

Let us now use the local results of Section 1.3 to prove the Γlim inf inequality.

Proof of Theorem 1.1. Let (σ ε , ϕ ε ) such that σ ε * σ and ϕ ε → ϕ as in the statement of the theorem. Without loss of generality, we can suppose that F ε (σ ε , ϕ ε ) < +∞.

Let Ω be as in the proof of Theorem 1.3 and let us define

χ = Γ -lim inf ε F ε (σ ε , ϕ ε ) and λ = β|σ| + H 1 Σ σ .
χ(A) ≥ µ(O i ∩ A) ≥ λ(O i ∩ A) = A ψ i dλ,
where

ψ i := 1 O i . Therefore Lemma 1.1 gives Γ -lim inf ε↓0 F ε (σ ε , ϕ ε ) = µ( Ω) ≥ λ( Ω) = β|σ|(Ω) + H 1 (Σ σ )
since sup i ψ i is the constant function 1.

Γ-limsup inequality

Let us prove the Γ-limsup inequality stated in Theorem 1.2. Recall that the latter consists in finding a sequence (σ ε , ϕ ε ) for any given couple (σ, ϕ)

∈ M(Ω, R 2 ) × L 1 (Ω) such that σ ε * σ, ϕ ε → ϕ in L 1 (Ω) and lim sup ε↓0 F ε (σ ε , ϕ ε ) ≤ E β (σ, ϕ).
(1.23)

When E β (σ, ϕ) = +∞ the inequality is valid for any sequence therefore by definition (1.5) we can assume σ = (m, τ, Σ) and ϕ = 1. In view of the results from 1.5. Γ-limsup inequality White [START_REF] White | Rectifiability of flat chains[END_REF], [START_REF] White | The deformation theorem for flat chains[END_REF] and Xia [Xia03] polyhedral vector measures are dense in energy and it is sufficient to consider vector measures of the form

σ = n i=1 (m i , τ i , Σ i ), (1.24)
where Σ i is a segment, m i ∈ R + is H 1 -a.e. constant and τ i is an orientation of Σ i for each i. We included in appendix A a proof of this result based on BV functions.

Without loss of generality we can suppose that for each couple of segments Σ i , M j , for i = j, the intersection Σ i ∩ M j is at most a point (called branching point) not belonging to the relative interior of Σ i and M j . We first produce the estimate (1.23) for σ concentrated on a single segment thus let us assume σ = me 1 H 1 (0, l) × {0}.

Notation: Let us fix the values

a ε :=    mβ ε 2 if β > 0 ε if β = 0 , b ε := ε ln 1 -η ε and r ε = max{ε, a ε }.
Let d ∞ (x, S) be the distance function from x to the set S ⊂ Ω relative to the infinity norm on R 2 and Q r (P ) = {x ∈ R 2 : d ∞ (x, P ) ≤ r} the square centered in P of size 2r and sides parallel to the axes. Introduce the sets

I aε := {x ∈ R 2 : d ∞ (x, [0, l] × {0}) ≤ a ε } ∪ Q rε (0, 0) ∪ Q rε (l, 0), I bε := {x ∈ R 2 : d ∞ (x, I aε ) ≤ b ε } \ I aε , I cε := {x ∈ R 2 : d ∞ (x, (I aε ∪ I bε )) ≤ ε} \ (I aε ∪ I bε ) , I dε := Ω \ (I aε ∪ I bε ∪ I cε ), Σ ε (t) := {(t, x 2 ) : |x 2 | ≤ r ε },
and define R ε = I aε \ (Q rε (0, 0) ∪ Q rε (l, 0)). l l B ε B ε B ε B ε Σ ε (r ε ) Σ ε (l -r ε ) Σ ε (r ε ) Σ ε (l -r ε ) R ε R ε Q rε (0, 0) Q rε (l, 0) Q rε (0, 0) Q rε (l, 0) I bε I cε I bε I cε
I aε = R ε ∪ (Q rε (0, 0) ∪ Q rε (l, 0)). Remark that supp(ρ ε ) = B(0, ε).
Costruction of σ ε : We build σ ε as a vector field supported on I aε . In particular we add together three different constructions performed respectively on R ε , Q rε (0, 0) and Q rε (l, 0). We construct σ ε on R ε in order to obtain the Γ-limsup inequality, on the other hand we are forced to modify such construction in a square neighborhood of each ending point of the segment to control ∇ • σ ε . As a matter of fact we need to verify that the piecewise definitions coincide on the sets Σ ε (r ε ) and Σ ε (lr ε ) which correspond to the interfaces between Q rε (0, 0) and R ε , and, R ε and Q rε (l, 0). Let r = r ε /ε and consider the problem

           ∆u = ±m (δ x 0 * ρ) , on Q r (0, 0), ∂u ∂ν = ±m H 1 (Σ) , on Σ ± = x ∈ R 2 : x 1 = ±1, |x 2 | ≤ mβ 2 , ∂u ∂ν = 0, otherwise on ∂Q r (0, 0) \ Σ ± .
(0, 0)

B 1 (0, 0) Σ - Σ + Q r (0, 0)
In the latter the set Σ + (resp. Σ -) is the image of the set Σ ε (r ε ) (resp. Σ ε (lr ε )) via the map

x -→ x ε , resp. x -→ x -(l, 0) ε .
Let u + be the solution relative to the problem in which every occurrence of ± is replaced by + and let u -defined accordingly. Then set

σ ε =              ∇u + (x/ε) ε , on Q rε (0, 0), m 2a ε e 1 , on R ε , ∇u -((x -(l, 0))/ε) ε , on Q rε (l, 0).
(1.25) Indeed, the Neumann Boundary conditions imposed for u + (resp. u -) on Σ + (resp. Σ -) ensure that the latter piecewise definition is continuous on Σ ε (r ε ) and Σ ε (lr ε ). By construction we have that ∇ • σ ε = m [(δ (0,0)δ (l,0) ) * ρ ε ] and σ ε * σ. Let us point out as well that there exists a constant c(α, m) such that c(α, m) :=

Qr ε (l,0) |σ ε | 2 dx = Qr ε (0,0) |σ ε | 2 dx = Qr(0,0) ∇u + (x) 2 dx = Qr(0,0) ∇u -(x) 2 dx.
(1.26)

Γ-limsup inequality

Costruction of ϕ ε : Most of the properties of ϕ ε are a consequence of the inequalities obtained in Theorem 1.5 and the structure of σ ε . On one hand we need ϕ ε to attain the lowest value possible on I aε in order to compensate the concentration of σ ε in this set, on the other, as shown in inequality (1.16), we need to provide the optimal profile for the transition from this low value to 1. For this reasons we are led to consider the following ordinary differential equation associated with the optimal transition

   w ε = 1 ε (1 -w ε ), w ε (0) = η.
(1.27)

Observe that w ε = 1 -(1 -η) exp -t ε
is the explicit solution of equation (1.27) and set

ϕ ε (x) :=          η, if x ∈ I aε , w ε (d ∞ (x, I aε )), if x ∈ I bε , d ∞ (x, I bε ) -ε + 1, if x ∈ I cε , 1,
otherwise.

(1.28)

Evaluation of F ε (σ ε , ϕ ε ): (case of a σ concentrated on a line segment)
We prove inequality (1.23) for the sequence we have produced. Since the sets I aε , I bε , I cε and I dε are disjoint we can split the energy as follows

F ε (σ ε , ϕ ε ) = F ε (σ ε , ϕ ε ; I aε ) + F ε (σ ε , ϕ ε ; I bε ) + F ε (σ ε , ϕ ε ; I cε ) + F ε (σ ε , ϕ ε ; I dε ) (1.29)
and evaluate each component individually. Since σ ε is null and ϕ ε is constant and equal to 1 in I dε we have that F ε (σ, ϕ ε ; I dε ) = 0. For the other components we strongly use the definitions in (1.25) and (1.28). First we split again the energy on the set I aε as following

F ε (σ ε , ϕ ε ; I aε ) = F ε (σ ε , ϕ ε ; R ε ) + F ε (σ ε , ϕ ε ; Q rε (0, 0)) + F ε (σ ε , ϕ ε ; Q rε (l, 0)).
Now identity (1.26) leads to the estimate

F ε (σ ε , ϕ ε ; Q rε (0, 0)) = F ε (σ ε , ϕ ε ; Q rε (l, 0)) = η 2 2ε c(β, m) + (1 -η) 2 2ε r 2 ε and F ε (σ ε , ϕ ε ; R ε ) = 1 2ε η 2 m 2a ε 2 + (1 -η) 2 2ε |R ε | ≤ (mη) 2 8εa 2 ε + 1 2ε 2a ε l.
Then passing to the limsup we obtain

lim sup ε↓0 F ε (σ ε , ϕ ε ; I aε ) ≤ mβl = mβH 1 ([0, l] × {0}).
(1.30)

To obtain the inequality on the sets I bε and I cε we are going to apply the coarea formula therefore let us observe that for both d ∞ (x, I aε ) and d ∞ (x, I bε ) there holds |∇d ∞ (x, •)| = 1 for a.e. x ∈ Ω and that there exist a constant k = k(β, m) such that Chapter 1. Affine cost function the level lines {d ∞ (x, •) = t} have H 1 length controlled by 2l + kt. In view of these remarks we obtain

F ε (σ ε , ϕ ε ; I bε ) = I bε ε 2 |∇ϕ ε | 2 + (1 -ϕ ε ) 2 2ε |∇d ∞ (x, I aε )| dx = bε 0 (1 -w ε (t)) 2 2ε + ε 2 |w ε (t)| 2 H 1 ({d ∞ (•, I aε ) = t}) dt ≤ (2l + kε) 1 2 (1 -w ε (t)) 2 bε 0 = l - kε 2 (1 -η) 2 -ε 2 --→ ε↓0 l = H 1 ([0, l] × {0}) (1.31)
and

F ε (σ ε , ϕ ε ; I cε ) = Ic ε ε 2 |∇ϕ ε | 2 + (1 -ϕ ε ) 2 2ε |∇d ∞ (x, I bε )| dx = ε 0 (1 -t + ε -1) 2 2ε + ε 2 H 1 ({d ∞ (•, I bε ∪ I aε ) = t}) dt ≤ (2l + kε) (t -ε) 3 6ε + ε 2 t ε 0 = (2l + kε) ε 2 2 3 --→ ε↓0 0. (1.32)
Finally adding up equations (1.29), (1.30), (1.31) and (1.32) we obtain lim sup

ε↓0 F ε (σ ε , ϕ ε ) ≤ (1 + β m) H 1 ([0, l] × {0}).
Case of a generic σ of the form (1.24): Let us call σ i ε , ϕ i ε the functions obtained above for each σ i = m i τ i H 1 Σ i and set

σ ε = n i=1 σ i ε , ϕ ε = min i ϕ i ε .
In view of the constraint, it holds

(µ + -µ -) = ∇ • σ = i ∇ • (m i τ i H 1 Σ i ) = i m i (δ P + i -δ P - i ),
where P + i and P - i are the starting and ending point of the segment Σ i according to its orientation τ i . Replacing each σ i with σ i ε we have

∇ • σ ε = i ∇ • σ i ε = i m i (δ P + i -δ P - i ) * ρ ε = (µ + -µ -) * ρ ε .
Thus σ ε satisfies constraint (1.1). We now prove inequality (1.23). The following inequality holds true

F ε (σ ε , ϕ ε ) = Ω 1 2ε | min i ϕ i ε | 2 | n i=1 σ i ε | 2 + ε 2 |∇(min i ϕ i ε )| 2 + (1 -min i ϕ i ε ) 2 2ε dx ≤ Ω 1 2ε | min i ϕ i ε | 2 | n i=1 σ i ε | 2 dx + n i=1 Ω ε 2 |∇ϕ i ε | 2 + (1 -ϕ i ε ) 2 2ε dx, (1.33)
1.6. Numerical Approximation therefore let us estimate the first integral in the latter. Observe that for ε sufficiently small we can assume that all the R i ε are pairwise disjoint thus we study the behavior in the squares. Let M i 1 , . . . , M im P be the segments meeting at a branching point P . For j = i 1 , . . . , i m P let us call Q r j ε (P ) the squared neighborhood of P relative to the segment M j as constructed previously. Let us recall that by definition ϕ ε is constant and equal to η on ∪ m P j=i 1 Q r j ε (P ) then we have the estimate

∪ m P j=i 1 (R j ε ∪Q r j ε (P )) ϕ 2 ε 2ε |σ ε | 2 dx = m P j=i1 R j ε ϕ 2 ε 2ε |σ ε | 2 dx + ∪ m P i=i 1 Q r j ε (P ) ϕ 2 ε 2ε | m P j=i1 σ j ε | 2 dx ≤ m P j=i1 R j ε ϕ 2 ε 2ε |σ ε | 2 dx + m P η 2 2ε m P j=i1 Q r j ε (P ) |σ j ε | 2 dx ≤ m P j=i1 (R j ε ∪Q r j ε (P )) 1 2ε |ϕ j ε | 2 |σ j ε | 2 dx + (m P -1)   im P j=i1 c(β, m j )   η 2 2ε c(m P ,β,mi 1 ,...,mi m P )ε
.

(1.34)

Applying inequality (1.34) on each branching point in equation (1.33) and recomposing the integral gives

lim sup ε↓0 F ε (σ ε , ϕ ε ) ≤ lim sup ε↓0 n i=1 F ε (σ i ε , ϕ i ε ) + n c(n, β, m i , . . . , m n )ε ≤ n i=1 (1 + β m i ) H 1 (Σ i ) = supp(σ) (1 + β m) dH 1 = E β (σ, 1)
which ends the proof.

Numerical Approximation

In this section we present numerical evidence of the Γ-convergence result we have shown in the setting of the Steiner Minimal Tree problem. Thus we consider µ + = δ x 0 and µ -= 1 N N j=1 δ x j for some points {x 0 , . . . , x N } ⊂ Ω. The first issue we address is how to impose the divergence constraint. To this aim it is convenient to introduce the Chapter 1. Affine cost function following notation

f ε = δ x 0 - 1 N N j=1 δ x j * ρ ε , G ε (σ, ϕ) =    Ω 1 2ε |ϕ| 2 |σ| 2 dx, if σ ∈ V ε and ϕ ∈ W ε , +∞, otherwise in L 2 (Ω, R 2 ), Λ ε (ϕ) =    Ω ε 2 |∇ϕ| 2 + (1 -ϕ) 2 2ε dx, if ϕ ∈ W ε , +∞, otherwise in L 1 (Ω).
(1.35)

Then let us observe that the following equality holds min

σ∈L 2 (Ω,R 2 ) G ε (σ, ϕ) = inf σ∈L 2 (Ω,R 2 ) sup u∈W 1,2 (Ω) Ω 1 2ε |ϕ| 2 |σ| 2 + u(∇ • σ -f ε ) dx .
By von Neumann's min-max Theorem [ABM14, Thm. 9.7.1] we can exchange inf and sup obtaining for each ε > 0 and

ϕ ∈ W ε min σ G ε (σ, ϕ) = sup u inf σ Ω 1 2ε |ϕ| 2 |σ| 2 -( ∇u, σ + uf ε ) dx = -min u Ω ε|∇u| 2 2|ϕ| 2 + uf ε dx = -min u G ε (u, ϕ),
with σ = ε∇u ϕ 2 , this naturally leads to the following alternate minimization problem: given an initial guess ϕ 0 we define

σ j := ε∇u j ϕ 2 j where u j := argmin G ε (u, ϕ j ), ϕ j+1 := argmin G ε (σ j , ϕ) + Λ ε (ϕ).
This formulation led to Algorithm 1. We define a circular domain Ω containing the points in S endowed with a uniform mesh and four values β, ε in , ε end and N iter and a gaussian convolution kernel ρ ε end in order to define f ε . We have implemented the algorithm in FREEFEM++ choosing for the discrete spaces for u and ϕ the space of piecewise polynomials of order 1. To validate Algorithm 1 we tested on the constraint given by four points defining a square inscribed in the unitary circumference, namely

x 0 = (- √ 2/2, - √ 2/2), x 1 = ( √ 2/2, - √ 2/2), x 2 = ( √ 2/2, √ 2/2) and x 4 = (- √ 2/2, √ 2/2).
Indeed, for such constraint, we can obtain an explicit solution which allows a visual comparison with the one obtained with Algorithm 1. As it is shown in Figure 1.3 the solution is far from being satisfactory. We think the failure of this procedure is due to the relation between the geometry of the space and the one of the solution itself. In particular, to obtain a good approximation it is necessary to refine the mesh where ϕ attains values close to zero but we observed that this restrains the process of approaching the solution. To overcame the problem we propose a modifications to Algorithm (1) to include a step of joint minimization. Let us consider a smooth diffeomorphism T : Ω → Ω and define

1.6. Numerical Approximation Algorithm 1 Alternate Minimization Require: S = {x 0 , . . . , x N }, ε in , ε end , N iter , β, index. function Alt. Min.(x 0 , . . . , x N , ε in , ε end , N iter , β, ρ) set f ε = (N δ x 0 -N i=1 δ x i ) * ρ ε end and ϕ 0 = 1 for j = 1, . . . , N iter do ε j = j-N iter N iter ε in - j N iter ε end φ ← L 1 -projection of ϕ 2 j-1 set u j := argmin G ε j (u, ϕ j-1 ) set σ j = ε j ∇u j φj-1 set ϕ j := argmin G ε j (σ j , ϕ) + Λ ε (ϕ) set ϕ j = max{η, ϕ j } end for end function return ϕ N iter , σ N iter .
ϕ T = ϕ • T (x), σ T = σ • T (x),
and the functional

F ε (T ) = G ε (σ • T, ϕ • T ) + Λ ε (ϕ • T ).
Let dF ε (Id) be the differential of the the functional F ε evaluated for T = Id. We represent dF as function V ∈ W 1,2 (Ω, Ω) by solving the elliptic problem

V, W W 1,2 = dF ε (Id), W W 1,2 for any test vector field W.
Let V be a solution to the latter problem, we perform a gradient descent in the direction -V . In Algorithm 2, we implemented this joint minimization step. As it is possible to remark from a visual comparison of modification to Algorithm 1. Let us observe that the optimization of the component 1.6. Numerical Approximation Λ ε defined in equation (1.35) is the one responsible for the length minimization of the support of σ. Therefore it is reasonable to look for a gradient descent in the component Λ ε . In Algorithm 3 we implement such procedure. This method enhances the length

S = {x 0 , . . . , x N }, ε in , ε end , N iter , β, index, N freq . function Joint Min.(x 0 , . . . , x N , ε in , ε end , N iter , β, ρ) set f ε = (N δ x 0 -N i=1 δ x i ) * ρ ε end and ϕ 0 = 1 for j = 1, . . . , N iter do ε j = j-N iter N iter ε in - j N iter ε end φ ← L 1 -projection of ϕ 2 j-1 set u j := argmin G ε j (u, ϕ j-1 ) set σ j = ε j ∇u j φj-1 set ϕ j := argmin G ε j (σ j , ϕ) + Λ ε (ϕ) if j%N freq == 0 & j ≥ index then solve V, W = dF ε j (Id), W set ϕ j = ϕ j (x -V ) end if set ϕ j = max{η, ϕ j } end for end function return ϕ N iter , σ N iter .
Algorithm 3 Length Minimization Require: S = {x 0 , . . . , x N }, ε in , ε end , N iter , β, index, N freq . function Length Min.(x 0 , . . . , x N , ε in , ε end , N iter , β, ρ) set f ε = (N δ x 0 -N i=1 δ x i ) * ρ ε end and ϕ 0 = 1 for j = 1, . . . , N iter do ε j = j-N iter N iter ε in - j N iter ε end φ ← L 1 -projection of ϕ 2 j-1 set u j := argmin G ε j (u, ϕ j-1 ) set σ j = ε j ∇u j φj-1 set ϕ j := argmin G ε j (σ j , ϕ) + Λ ε (ϕ) if j%N freq == 0 & j ≥ index then solve V, W = dΛ ε j (Id), W set ϕ j = ϕ j (x + T ) end if set ϕ j = max{η, ϕ j } end for end function return ϕ N iter , σ N iter .
minimization process since but has the drawback is that displacing ϕ and σ in the direction -dΛ we could loose the divergence constraint. To avoid such eventuality we perform several steps of Alternate Minimization after the displacement. In the next figures we show the graphs obtained for the couple (σ N iter , ϕ N iter ) via Algorithm 3 with the choices β = 0.05, ε in = 0.5, ε end = 0.05, β = 0.05, N iter = 500 and index = 300. We have chosen to make simulations for points located on the vertices of regular polygons of respectively 3, 4, 5 and 6 vertices. A direct visual comparison between the obtained results in Figure 1.6 and the exact solutions in Figure 1.7. Chapter 2

Multidimensional case 2.1 Introduction

The content of this and the following chapter are the argument of the published paper [START_REF] Chambolle | Variational approximation of size-mass energies for k-dimensional currents[END_REF]. We generalize the result obtained in the previous chapter to any ambient space R n . Indeed, since n > 2 we cannot take advantage of the localization procedure described previously, namely divergence free vector valued measures may not be regarded as the rotated gradient of a BV function. For this matter we need to introduce different techniques from the previous ones. Furthermore for scaling issues we need to consider a p-laplacian energy rather than the elliptic one in the Modica-Mortola component of the functional similarly to [START_REF] Ghiraldin | Variational approximation of a functional of Mumford-Shah type in codimension higher than one[END_REF]. Here we take advantage of a result from White [START_REF] White | Rectifiability of flat chains[END_REF][START_REF] White | The deformation theorem for flat chains[END_REF] to show that if the family of functionals is equibounded in energy, then up to a subsequence we can extract a sequence of vector measures converging to a 1-rectifiable vector measure. Furthermore the result is based on the study of a dimension reduced problem which is studied in the appendix. This reduced dimension problem has some analogies with the functional studied in [START_REF] Bouchitté | Transitions de phases avec un potentiel dégénéré à l'infini, application à l'équilibre de petites gouttes[END_REF] in the case of droplets equilibrium featuring measures with fixed total mass which concentrate on atoms.

Let us define the approximating family of functionals precisely. Again let µ + and µ -be two probability measures supported on a countable number of points. Let ρ : R n → R + be a classical radial mollifier such that supp ρ ⊂ B 1 (0) and B 1 (0) ρ = 1. For ε > 0, we set ρ ε = ε -n ρ(•/ε). Consider vector fields satisfying equation

∇ • σ ε = (µ + -µ -) * ρ ε in D (R n ). (2.1)
We also consider the functions

ϕ ∈ W 1,p (Ω, [η, 1]) such that ϕ ≡ 1 on ∂Ω, where η = η(ε) satisfies η = β ε n (2.2)
for some β ∈ R + . We denote by X ε (Ω) the set of pairs (σ, ϕ) such that ϕ is as stated above and σ ∈ L 1 (Ω, R n ) satisfies equation (2.1). This set is naturally embedded in

M(Ω, R n ) × L 2 (Ω). For (σ, ϕ) ∈ M(Ω, R n ) × L 2 (Ω) and p > n -1 we set F ε,β (σ, ϕ; Ω) :=      Ω ε p-n+1 |∇ϕ| p + (1 -ϕ) 2 ε n-1 + ϕ|σ| 2 ε dx, if (σ, ϕ) ∈ X ε (Ω), +∞ , otherwise.
(2.3)

Chapter 2. Multidimensional case Let X be the subset of M(Ω, R n ) × L 2 (Ω) consisting of those couples (σ, ϕ) such that ϕ ≡ 1 and σ = (m, τ, Σ) satisfies the constraint ∇ • σ ε = µ + -µ - in D (R n ).
(2.4)

Given any sequence ε = (ε i ) i∈N of positive numbers such that ε i ↓ 0, we show that F ε,β family of functionals Γ-converges to

E β (σ, ϕ; Ω) =          Σ∩Ω h β (m(x)) dH 1 (x), if (σ, ϕ) ∈ X and σ = m τ H 1 Σ, +∞ , otherwise.
(2.5)

The function h β : R + → R + (introduced and studied in Appendix B) is the minimum value of some optimization problem depending on β and on the codimension n -1 (we note h d β , with d = nk in the general case 1 ≤ k ≤ n -1). In particular we prove that h β is lower semicontinuous, subadditive, increasing, h β (0) = 0 and that there exists some c > 0 such that

1 c ≤ h β (m) 1 + √ β m ≤ c for m > 0.
(2.6)

The Γ-convergence holds for the topology of the weak- * convergence for the sequence of measures (σ ε ) and for the strong L 2 convergence for the phase field (ϕ ε ). For a sequence

(σ ε , ϕ ε ) we write (σ ε , ϕ ε ) → (σ, ϕ) if σ ε * σ and ϕ ε -ϕ L 2 → 0.
In the sequel we first establish that the sequence of functionals (F ε,β ) ε is coercive with respect to this topology.

Theorem 2.1 (Equicoercivity). Assume that β > 0. For any sequence

(σ ε , ϕ ε ) ⊂ M(Ω, R n ) × L 2 (Ω) with ε ↓ 0, such that F ε,β (σ ε , ϕ ε ; Ω) ≤ F 0 < +∞, there exists σ ∈ M(Ω, R n ) such that, up to a subsequence, (σ ε , ϕ ε ) → (σ, 1) ∈ X.
Then we prove the Γ-liminf inequality

Theorem 2.2 (Γ-lim inf inequality). Assume that β ≥ 0. For any sequence (σ ε , ϕ ε ) ⊂ M(Ω, R n ) × L 2 (Ω) that converges to (σ, ϕ) ∈ M(Ω, R n ) × L 2 (Ω) as ε ↓ 0 it holds lim inf ε↓0 F ε,β (σ ε , ϕ ε ; Ω) ≥ E β (σ, ϕ; Ω).
We also establish the corresponding Γ-limsup inequality

Theorem 2.3 (Γ-lim sup inequality). Assume that β ≥ 0. For any (σ, ϕ) ∈ M(Ω, R n )× L 2 (Ω) there exists a sequence (σ ε , ϕ ε ) ⊂ M(Ω, R n ) × L 2 (Ω) such that (σ ε , ϕ ε ) ε↓0 -→ (σ, ϕ) in M(Ω, R n ) × L 2 (Ω) and lim sup ε↓0 F ε,β (σ ε , ϕ ε ; Ω) ≤ E β (σ, ϕ; Ω).

Reduced problem results in dimension nk

Notice that the coercivity of the family of functionals only holds in the case β > 0. However, as β ↓ 0 we have the important phenomena:

h β β↓0 -→ c1 (0,+∞) pointwise,
for some c > 0. As a consequence (2.5) is an approximation of cH 1 (Σ) for β > 0 small. In the first section of this chapter we anticipate the optimization problem defining the cost function h d β and some results which are studied in the Appendix B. This problem is studied independently as it is useful to obtain similar results for k-currents replacing vector valued measures. This extension is studied in the following chapter.

Reduced problem results in dimension nk

This section is devoted to introducing some notation and results corresponding to the case k = 1. In the sequel, these results are used to describe the energetical behaviour of the (nk)-dimensional slices of the configuration (σ ε , ϕ ε ). We postpone the proofs to Appendix B.3, B.4 and B.5. We set d = nk, p > d and consider ε to be a sequence such that ε ↓ 0. Let B r (0) ⊂ R d be the ball of radius r centered in the origin. We consider the functional

G ε,β (ϑ, ϕ; B r ) := Br ε p-d |∇ϕ| p + (1 -ϕ) 2 ε d + ϕ|ϑ| 2 ε dx (2.7)
where ϕ ∈ W 1,p (B r ) is constrained to satisfy the lower bound ϕ ≥ β ε d+1 =: η and ϑ ∈ L 2 (B r ) is such that supp(ϑ) ⊂ B r with 0 < r < r, ϑ 1 = m. This leads to define the set

Y ε,β (m, r, r) = (ϑ, ϕ) ∈ L 2 (B r ) × W 1,p (B r , [η, 1]) : ϑ 1 = m and supp(ϑ) ⊂ B r ,
and the optimization problem

h d ε,β (m, r, r) = inf Y ε,β (m,r,r) G ε,β (ϑ, ϕ; B r ). (2.8) Let h d β : [0, +∞) -→ R + be defined as h d β (m) =          min r>0 β m 2 ω d rd + ω d rd + (d -1) ω d q d ∞ (0, r) , for m > 0, 0, for m = 0, (2.9) with q d ∞ (ξ, r) := inf +∞ r t d-1 |v | p + (1 -v) 2 dt : v(r) = ξ and lim t→+∞ v(t) = 1 ,
(2.10) for r > 0, ξ ≥ 0. For a graph of the profile v realizing the infimum in the latter see Figure B.1. We have the following results

Chapter 2. Multidimensional case Proposition 2.1. For any r > r > 0, it holds

lim inf ε↓0 h d ε,β (m, r, r) ≥ h d β (m).
(

2.11)

There exists a uniform constant κ := κ(d, p) such that

h d β (m) ≥ κ for every m > 0.
(2.12)

Proposition 2.2. For fixed m > 0 let r * be the minimizing radius in the definition of h d β (m) (2.9). For any δ > 0 and ε small enough there exist a function ϑ ε = c1 Br * ε with c > 0 such that Br ϑ ε = m and a nondecreasing radial function

ϕ ε : B r → [η, 1] such that ϕ ε (0) = η, ϕ ε = 1 on ∂B r and G ε,β (ϑ ε , ϕ ε ; B r ) ≤ h d β (m) + δ.
(2.13)

Proposition 2.3. The function h d β is continuous in (0, +∞), increasing, sub-additive and h d β (0) = 0.

Compactness

We prove the compactness Theorem 2.1 for the family of functionals

(F ε,β ) ε . Let us consider a family of functions (σ ε , ϕ ε ) ε↓0 , such that (σ ε , ϕ ε ) ∈ X ε (Ω) and F ε,β (σ ε , ϕ ε ; Ω) ≤ F 0 . (2.14)
As a first step we prove:

Lemma 2.1. Assume β > 0. There exists C ≥ 0, depending only on Ω, F 0 and β such that

Ω |σ ε | ≤ C, ∀ ε. (2.15)
As a consequence there exist a positive Radon measure µ ∈ (R n , R + ) supported in Ω and a vectorial Radon measure σ ∈ M(Ω, R n ) with ∇ • σ = a j δ x j and |σ| ≤ µ such that up to a subsequence

ϕ ε → 1 in L 2 (Ω), |σ ε | * µ in M(R n ), σ ε * σ in M(R n , R n ).
Proof. We divide the proof into three steps.

Step 1. We start by proving the uniform bound (2.15). Let λ ∈ (0, 1] and let

Ω λ := {x ∈ Ω : ϕ ε > λ} .
Being σ ε square integrable we identify the measure σ ε with its density with respect to L n . Therefore splitting the total variation of σ ε , we write

|σ ε |(Ω) = Ω |σ ε | dx = Ω λ |σ ε | dx + Ω\Ω λ |σ ε | dx.

Compactness

We estimate each term separately. By the Cauchy-Schwarz inequality we have

Ω λ |σ ε | ≤ Ω λ ϕ ε |σ ε | 2 ε 1/2 Ω λ ε ϕ ε 1/2 . Since λ < ϕ ε ≤ 1 on Ω λ and Ω λ (ϕ ε |σ ε | 2 )/(ε) dx is bounded by F ε,β (σ ε , ϕ ε ; Ω), from
the previous inequality we get

Ω λ |σ ε | ≤ Ω λ ϕ ε |σ ε | 2 ε 1/2 |Ω|ε λ ≤ |Ω| ε F 0 λ .
Next, in Ω \ Ω λ , by the Young inequality, we have 2

Ω\Ω λ |σ ε | ≤ Ω\Ω λ ϕ ε |σ ε | 2 ε + Ω\Ω λ ε ϕ ε . Using ϕ ε ≥ η(ε), η/ε n = a and (1 -λ) 2 ≤ (1 -ϕ ε ) 2 in Ω \ Ω λ , we obtain Ω\Ω λ |σ ε | ≤ 1 2 Ω ϕ ε |σ ε | 2 ε + ε n 2 η (1 -λ) 2 Ω (1 -ϕ ε ) 2 ε n-1 ≤ F 0 2 + F 0 2 β (1 -λ) 2 . Hence |σ ε |(Ω) ≤ F 0 2 + F 0 2 β (1 -λ) 2 + |Ω| ε F 0 λ .
As β > 0, this yields (2.15).

Step 2. We easily see from

Ω (1 -ϕ ε ) 2 ≤ F 0 ε n-1 that ϕ ε → 1 in L 2 (Ω) as ε ↓ 0.
Step 3. The existence of the Radon measures µ and σ such that, up to extraction, |σ ε | * µ and σ ε * σ follows from (2.15). The properties on the support of µ, on the divergence of σ and the fact that |σ| ≤ µ follow from the respective properties of σ ε .

We have just showed that the limit σ of a family (σ ε , ϕ ε ) ε equibounded in energy is bounded in mass. In what follows, we assume β ≥ 0 and that σ ε is bounded in mass. We show that the limiting σ is rectifiable.

Proposition 2.4. Assume β ≥ 0 and that the conclusions of Lemma 2.1 hold true. There exists a Borel subset Σ with finite length and a Borel measurable function τ : Σ → S n-1 such that σ = τ |σ| Σ. Moreover, we have the following estimate,

H 1 (Σ) ≤ C * F 0 ,
where the constant C * ≥ 0 only depends on d and p.

This proposition together with Lemma 2.1 and Theorem 0.5 leads to Proposition 2.5. σ is a 1-rectifiable vector measure and in particular Σ is a countably H 1 -rectifiable set.

Chapter 2. Multidimensional case The latter ensures that the limit couple (σ, 1) belongs to X and concludes the proof of Theorem 2.1. We now establish Proposition 2.4 Sketch of the proof: We first define Σ. Then we show in Lemma 2.3 that for x ∈ Σ, we have lim inf ε↓0 F ε,β (σ ε , ϕ ε ; B(x, r j )) ≥ κr j for a sequence of radii r j ↓ 0 and κ > 0. The proof of the lemma is based on slicing and on the results of Appendix B. The proposition then follows from an application of the Besicovitch covering theorem.

First we introduce the Borel set

Σ := x ∈ Ω : ∀r > 0, |σ|(B r (x)) > 0 and ∃τ = τ (x) ∈ S n-1 such that τ = lim r↓0 σ(Br(x)) |σ|(Br(x)) .
We observe that by Besicovitch derivation theorem,

σ = τ |σ| Σ.
Next we fix θ ∈ (0, 1/4 n ) and define

Γ := x ∈ Σ : ∃ r 0 > 0 such that, |σ|(B r/4 (x)) |σ|(B r (x)) ≤ θ for every r ∈ (0, r 0 ] .
We show that this set is |σ|-negligible.

Lemma 2.2. We have |σ|(Γ) = 0.

Proof. Let x ∈ Γ. Applying the inequality |σ|(B r/4 (x)) ≤ θ|σ|(B r (x)) with r = r k = 4 -k r 0 , k ≥ 0, we get |σ|(B r k ) ≤ θ k |σ|(B r 0 )
. Hence there exists C ≥ 0 such that |σ|(B r (x)) ≤ Cr (ln 1/θ)/(ln 4) .

Letting λ = (ln 1 θ )/(ln 4), we have by assumption λ > n. Therefore, for every ξ > 0 there exists r ξ = r ξ (x) ∈ (0, 1) such that

|σ|(B r ξ (x)) ≤ ξ|B r ξ (x)|. Now, for R > 0, we cover Γ ∩ B R with balls of the form B r ξ (x) (x). Using Besicovitch covering theorem, we have Γ ∩ B R ⊂ ∪ N (n) j=1 B j where N (n) only depends on n and each B j is a (finite or countable) disjoint union of balls of the form B r ξ (x k ) (x k ). Then we get |σ|(Γ ∩ B R ) ≤ N (n) j=1 |σ|(B j ) ≤ N (n)ξ|B j | ≤ N (n)|B R+1 |ξ.
Sending ξ to 0 and then R to ∞, we obtain |σ|(Γ) = 0.

Set Σ := Σ \ Γ, from Lemma 2.2, we have σ = τ |σ| Σ. Recall that S = supp µ + ∪ supp µ -. Lemma 2.3. For every x ∈ Σ \ S , there exists a sequence (r j ) = (r j (x)) ⊂ (0, 1) with

Compactness

r j ↓ 0 such that lim inf ε↓0 F ε,β (σ ε , ϕ ε ; B(x, r j )) ≥ √ 2 κ r j ,
where κ is the constant of Proposition 2.1.

Proof. Let x ∈ Σ \ S . Without loss of generality, we assume x = 0 and τ (x) = e 1 . Let ξ > 0 be a small parameter to be fixed later. From the definition of Σ, there exists a sequence (r j ) = (r j (x)) ⊂ (0, d(x, S )) such that for every j ≥ 0,

σ(B r j ) • e 1 ≥ (1 -ξ)|σ|(B r j ) and |σ|(B r j /4 ) ≥ θ|σ|(B r j ).
(2.16)

Let us fix j ≥ 0 and set, to simplify the notation, r = r j and r * = r/ √ 2. Recall the notation x = (x 1 , x ) ∈ R × R n-1 and define the cylinder (2.17) From (2.16), we see that m > 0 for ξ small enough. Indeed, we have

C r * := {x : |x 1 | ≤ r * and |x | ≤ r * } so that C r * ⊂ B r and B r/4 ⊂ C r * /2 , as shown in figure 2.1. Let χ ∈ C ∞ c (R n-1 , [0, 1]) be a radial cut-off function such that χ(x ) = 1 if |x | ≤ 1 2 and χ(x ) = 0 for |x | ≥ 3 4 . Then, we note χ r * (x ) = χ(x /r * ) and for s ∈ [-r, r], we set ∀s ∈ [-r, r], g ε (s) := e 1 • B r * σ ε (s, x ) χ r * (x ) dx . Since σ ε is divergence free, e 1 • σ ε (•,
(1 -ξ)|σ|(B r ) ≤ 2rm + σ(B r ) • e 1 = Br 1 -χ r * (x )1 [-r,r] dσ(s, x ) • e 1 ≤ 2rm + Br 1 -χ r * (x )1 [-r,r] d|σ|(s, x ) ≤ 2rm + |σ|(B r ) - Br χ r * (x )1 [-r,r] d|σ|(s, x ).
Since by construction χ r * (x )1 [-r,r] ≥ 1 B r/4 , using the second inequality of (2.16), we have

m ≥ 1 2r (θ -ξ)|σ|(B r ) > 0,
for ξ small enough. Similarly, denoting Π : R n → R n-1 , (t, x ) → x the orthogonal projection onto the last (n -1) coordinates, we deduce again from (2.16) that

|Πσ|(C r * ) ≤ √ ξ m m -ξ 2r.
(2.18) Now, for ε small enough, we have ∇ • σ ε = 0 in C r * . Using this, we have for almost every s, t ∈ [-r, r], with s < t,

g ε (t) -g ε (s) = t s B r * σ ε (x , h) • ∇ χ r * (x ) dx dh.
Integrating in s over (-r, r), we get for almost every t ∈ [-r, r],

g ε (t) -ḡε = 1 2r (-r,r)×B r * φ t (h, x ) • σ ε (h, x ) dx dh with φ t (h, x ) = (h + r) ∇ χ r * (x ) if h < t, (h -r) ∇ χ r * (x ) if h > t.
We deduce the following convergence

g ε (t) -m ε↓0 -→ 1 2r (-r,r)×B r * φ t (h, x ) • dσ(h, x ) (2.19)
in the L 1 (-r, r) topology. Using (2.18), we see that the above right hand side is bounded by c √ ξ m-ξ m. Taking into account (2.18) and the continuity of g ε , we conclude that lim inf

ε↓0 g ε (t) ≥ 1 -c √ ξ m -ξ m for t ∈ [-r, r].

Γ-liminf inequality

Next, by decomposing the integral we have

F ε,β (σ ε , ϕ ε ; B r ) ≥ r -r B r * ε p-n+1 |∇ϕ ε | p + (1 -ϕ ε ) 2 ε n-1 + ϕ ε |σ ε | 2 ε dx dt ≥ r -r B r * ε p-n+1 |∇ϕ ε | p + (1 -ϕ ε ) 2 ε n-1 + ϕ ε |χ r * (x )σ ε | 2 ε dx dt.
(2.20)

Let us set ϑ t ε (x ) := |χ r * (x )σ ε (t, x )|. By construction ϑ t ε has the properties:

• ϑ t ε ∈ L 1 (B r * ), • lim inf ε↓0 B r * ϑ t ε (x ) dx ≥ lim inf ε↓0 g ε (t) ≥ 1 -c √ ξ m-ξ m = m > 0,
• supp(ϑ t ε ) ⊂ B r with r := 3 4 r * < r * . By definition of the minimization problem introduced in Section 2.2, we have

F ε,β (σ ε , ϕ ε ; B r ) ≥ r -r inf (ϑ,ϕ)∈Y ε,β ( m,r,r) G ε,β (ϑ, ϕ; B r ) dt = r -r h n-1 ε,β ( m, r, r) dt.
(2.21) Taking the infimum limit, by Fatou's lemma and equation (2.12) of Proposition 2.1 we get

lim inf ε↓0 F ε,β (σ ε , ϕ ε ; B r ) ≥ r -r lim inf ε↓0 h n-1 ε,β ( m, r, r) dt ≥ 2 r κ.
The latter holds for almost every r ∈ [(1-ξ)r * , r * ] and eventually, since the r * = r/ √ 2, we conclude lim inf

ε↓0 F ε,β (σ ε , ϕ ε ; B r ) ≥ √ 2 κ r.
The proof of Proposition 2.4 is then obtained via the Besicovitch covering theorem [START_REF] Evans | Measure theory and fine properties of functions[END_REF].

Γ-liminf inequality

In this section we prove the Γlim inf inequality stated in Theorem 2.2.

Proof of Theorem 2.2. With no loss of generality we assume that lim inf ε↓0 F ε,β (σ ε , ϕ ε ) < +∞ otherwise the inequality is trivial. For a Borel set A ⊂ Ω, we define

H(A) := lim inf ε↓0 F ε,β (σ ε , ϕ ε ; A),
so that H is a subadditive set function. By assumption, the limit measure σ is 1rectifiable; we write σ = m τ H 1 Σ. Furthermore we can assume σ to be compactly supported in Ω. Consider a convex open set Ω 0 such that supp(∇•σ) = S ⊂⊂ Ω 0 ⊂⊂ Ω Chapter 2. Multidimensional case and let f := [0, 1] × R n → R n be a smooth homotopy of the indentity map on R n onto a contraction of Ω into Ω 0 such that f (t, •) restricted to Ω 0 is the identity map, for any

t ∈ [0, 1]. Let σ t = f (t, •) σ, indeed lim inf t↓0 F (σ t , 1) ≥ F (σ, 1) as σ t * σ. Further ∇ • σ t = ∇ • σ since h(t, •) is the identity on S . Now we claim that lim inf r↓0 H B(x, r) 2r ≥ h β (m(x)) for H 1 -almost every x ∈ Σ. (2.22)
Let us fix λ ≥ 1 and let us note h β,λ (t) := min(h β (t), λ). We then introduce the Radon measure

H λ (A) := Σ∩A h β,λ (m) dH 1 .
Now, let δ ∈ (0, 1). Assuming that (2.22) holds true, there exists Σ 0 ⊂ Σ with H 1 (Σ\Σ 0 ) = 0 such that for every x ∈ Σ 0 , there exists r 0 (x) > 0 with

(1 + δ)H B(x, r) ≥ 2rh β,λ (m(x))
for every r ∈ (0, r 0 (x)).

By the Besicovitch differentiation Theorem, there exists Σ 1 ⊂ Σ with H 1 (Σ\Σ 1 ) = 0 such that for every x ∈ Σ 1 , there exists r 1 (x) > 0 with

(1 + δ)2rh β (m(x)) ≥ H λ B(x, r) for every r ∈ (0, r 1 (x)).
We consider the familly B of closed balls B(x, r) with x ∈ Σ 0 ∩ Σ 1 and 0 < r < min(r 0 (x), r 1 (x)) and we apply the Vitali-Besicovitch covering theorem [AFP00, Theorem 2.19] to the family B and to the Radon measure H λ . We obtain a disjoint family of closed balls B ⊂ B such that

H λ (Ω) = H λ (Σ) = B(x,r)∈B H λ B(x, r) ≤ (1 + δ) 2 B(x,r)∈B H B(x, r) ≤ (1 + δ) 2 H(Ω).
Sending λ to infinity and then δ to 0, we get the lower bound H(Ω) ≥ Σ h β (m) dH 1 which proves the theorem.

Let us now establish the claim (2.22). Since σ is a rectifiable measure, we have for H 1 -almost every x ∈ Σ and for every ϕ

∈ C c (R n ), 1 2r ϕ(x + ry) d|σ|(y) r↓0 -→ m(x) R ϕ(tτ (x)) dt (2.23) and 1 2r B(x,r)∩Σ |τ (y) -τ (x)| d|σ|(y) r↓0 -→ 0.
(2.24) Let x ∈ Σ \ S be such a point. Without loss of generality, we assume x = 0, τ (0) = e 1 and m := m(0) > 0. Let δ ∈ (0, 1). Our goal is to establish a precise lower bound for F ε,β (σ ε , ϕ ε ; C) where C is a cylinder of the form

C δ r := {x ∈ R n : |x 1 | < δr, |x | < r} .

Γ-liminf inequality

For this we proceed as in the proof of Lemma 2.3, here, the rectifiability of σ simplifies the argument. Let

χ δ ∈ C ∞ c (R n-1 , [0, 1]) be a radial cut-off function with χ δ (x ) = 1 if |x | ≤ δ/2, χ δ (x ) = 0 if |x | ≥ δ.
For ε > 0 and r ∈ (0, d(0, ∂Ω)), we define for s ∈ (-r, r),

g δ,r ε (s) := e 1 • R n-1 σ ε (s, x ) χ δ (x /r) dx .
We also introduce the mean value

g δ,r ε := 1 2r r -r g δ,r ε (s) ds.
From (2.23), we have for r > 0 small enough,

g δ,r 0 := 1 2r r -r e 1 • R n-1 σ ε (s, x ) χ δ (x /r) dx ds ≥ (1 -δ)m.
For such r > 0, we deduce from σ ε * σ that for ε > 0 small enough

g δ,r ε := 1 2r r -r g δ,r ε (s) ds ≥ (1 -2δ)m. (2.25)
We study the variation of g δ,r ε (s). Using ∇ • σ ε = 0 in C δ r , we compute as in the proof of Lemma 2.3,

g δ,r ε (t) -g δ,r ε = 1 2r (-r,r)×B δr φ t (x , h) • σ ε (x , h) dx dh with φ t (h, x ) = (h + r) ∇ χ δ (x /r) if h < t, (h -r) ∇ χ δ (x /r) if h > t.
Using again the convergence σ ε * σ, we deduce

g δ,r ε (t) -g δ,r ε ε↓0 -→ 1 2r (-r,r)×B δr φ t (x , h) • dσ(x , h),
in L 1 (-r, r). Now, since e 1 • ∇ χ δ ≡ 0, we deduce from (2.24) that the right hand side goes to 0 as r ↓ 0. Hence, for r > 0 small enough,

1 2r (-r,r)×B δr φ t (x , h) • σ(x , h) dx dh ≤ δm.
Using (2.25), we conclude that for r > 0 small enough and then for ε > 0 small enough, we have

g δ,r ε (t) ≥ (1 -3δ)m, for a.e. t ∈ (-r, r).
By definition of the reduced dimension problem, we conclude that

F ε,β (σ ε , ϕ ε ; C δ r ) ≥ 2rh n-1 ε,β ((1 -3δ)m) .
Sending ε ↓ 0, we obtain

H(C δ r ) ≥ 2rh n-1 β ((1 -3δ)m) .
We notice that H(B √ 1+δ 2 r ) ≥ H(C δ r ). Recall that for the case n -1 we omit the superscript in the definition of h, thus dividing by 2 √ 1 + δ 2 r and taking the liminf as r ↓ 0, we get

lim inf r↓0 H(B √ 1+δ 2 r ) 2 √ 1 + δ 2 r ≥ h β ((1 -3δ)m) √ 1 + δ 2 .
Sending δ to 0, we get (2.22) by lower semi-continuity of h β .

Γ-limsup inequality

Proof of Theorem 2.3. Let us suppose F (σ, ϕ; Ω) < +∞, so that in particular ϕ ≡ 1. From Xia [START_REF] Xia | Interior regularity of optimal transport paths[END_REF], we can assume σ to be supported on a finite union of compact segments and to have constant multiplicity on each of them, namely polyhedral vector measures are dense in energy. We first construct a recovery sequence for a measure σ concentrated on a segment with constant multiplicity. Then we show how to deal with the case of a polyhedral vector measures.

Step 1. (σ concentrated on a segment.) Assume that σ is supported on the segment I = [0, L] × {0} and writes as m e 1 H 1 I . Consider m constant so that ∇ • σ = m(δ (0,0)δ (L,0) ) and

E β (σ, 1; Ω) = h β (m) H 1 (I) = L h β (m).
For δ > 0 fixed, we consider the profiles

ϕ ε (t) :=          η, for 0 ≤ t ≤ r * ε, v δ t ε , for r * ε ≤ t ≤ r, 1 for r ≤ t,
and

ϑ ε = m χ B r * ε (x ) ω n-1 (εr * ) n-1
with r * and v δ , defined in Proposition 2.2 with d = n -1. Assume r * ≥ 1 and let d(x, I) be the distance function from the segment I and introduce the sets

I r * ε := {x ∈ Ω : d(x, I) ≤ r * ε} , and
I r := {x ∈ Ω : d(x, I) ≤ r} . Set ϕ ε (x) = ϕ ε (d(x, I)) and σ 1 ε = (mH 1 I) * ρ ε
, where ρ ε is the mollifier of equation (2.1). We first construct the vector measures

σ 1 ε = σ 1 ε e 1 and σ 2 ε (x 1 , x ) = ϑ ε (|x |) e 1 .
Alternatively, σ 2 ε = σ * ρε for the choice ρε (x 1 , x ) = χ B r * ε (x )/ ω n-1 (εr * ) n-1 . Let us highlight some properties of σ 1 ε and σ 2 ε . Both vector measures are radial in x , with an abuse of notation we denote σ

1 ε (x 1 , s) = σ 1 ε (x 1 , |x |). Since,
| = σ 1 ε and |σ 2 ε | = ϑ ε . Furthermore for any x 1 , it holds {x 1 }×B r * ε σ 1 ε (x 1 , x ) -ϑ ε (x ) dx = 0. (2.26)
We construct σ ε by interpolating between σ 1 ε and σ 2 ε . To this aim consider a cutoff function

ζ ε : R → R + satisfying ζ ε (t) = 1 for t ≤ r * ε or t ≥ L -r * ε, ζ ε (t) = 0 for 2 r * ε ≤ t ≤ L -2 r * ε,
and

|ζ ε | ≤ 1 r * ε
and define σ ε component-wise as

     σ 3 ε • e 1 = 0, σ 3 ε • e i (x 1 , x ) = -ζ ε (x 1 ) x i |x | n-1 |x | 0 s n-2 σ 1 ε (x 1 , s) -ϑ ε (s) ds, for i = 2, . . . , n.
The integral corresponds to the difference of the fluxes of σ 1 ε and σ 2 ε through the (n-1)dimensional disk {x 1 } × B . For σ 3 ε we have the following

∇ • σ 3 ε = -ζ ε (x 1 ) n i=2 1 |x | n-1 - (n -1)x 2 i |x | n+1 |x | 0 s n-2 σ 1 ε (x 1 , s) -ϑ ε (s) ds + x 2 i |x | 2 σ 1 ε (x 1 , |x |) -ϑ ε (|x |) = -ζ ε (x 1 ) σ 1 ε (x 1 , |x |) -ϑ ε (|x |) . (2.27) Let σ ε = ζ ε σ 1 ε + (1 -ζ ε ) σ 2 ε + σ 3 ε .
In force of equation (2.27) and from the construction of σ 1 ε , σ 2 ε and ζ ε we have

∇ • σ ε = ∇ • (ζ ε σ 1 ε ) + ∇ • (1 -ζ ε )σ 2 ε + ∇ • σ 3 ε = ζ ε ∇ • σ 1 ε + ζ ε (σ 1 ε -ϑ ε ) + ∇ • σ 3 ε = ζ ε ∇ • σ 1 ε = ∇ • (σ * ρ ε ).
In addition for any (x 1 , x ) such that |x | ≥ r * ε from (2.26) we derive

σ 3 ε • e i (x 1 , x ) = -ζ ε (x 1 ) x i |x | n-2 |x | 0 s n-1 σ 1 ε (x 1 , s) -ϑ ε (s) ds = 0 which justifies supp(σ ε ) ⊂ I r * ε . Let us now prove lim sup ε↓0 F ε,β (σ ε , ϕ ε ; Ω) ≤ L h β (m) + Cδ.
We split Ω as the union of

Ω \ I r , C r,ε := I r ∩ [2 ε, L -2 ε] × R n-1 and D ε and D ε , as show in figure 2.2, where D ε = {x 1 ≤ 2 r * ε} ∩ I r * ε and D ε = {x 1 ≥ L -2 r * ε} ∩ I r * ε .
On Ω \ I r we notice that σ ε = 0 and ϕ ε = 1 therefore Observe that

F ε,β (σ ε , ϕ ε ; Ω \ I r ) = 0. Chapter 2. Multidimensional case Ω O L εr * 2εr * L -2εr * L -εr * D ε D ε I r * ε I r
|D ε | = |D ε | = Cε n , then we have the upper bound Dε |σ ε | 2 dx ≤ 2 m 2 r 2 * ε n-2 B 1 ρ 2 dx + C .
Taking into consideration this estimate we obtain

F ε,β (σ ε , ϕ ε ; D ε ) = F ε,β (σ ε , ϕ ε ; D ε ) ≤ (1 -η) 2 ε n-1 L n (D ε ) + 2 m 2 r 2 * η ε n-2 .
(2.28)

Finally on C r,ε both σ ε and ϕ ε are independent of x 1 and are radial in x then by Fubini's theorem and Proposition 2.2 we get

F ε,β (σ ε , ϕ ε ; C r,ε ) = L-2 εr * 2 εr * B r G ε,β (ϑ ε , ϕ ε ) ≤ L (h β (m) + C δ).
Adding all together gives the desired estimate. It remains to discuss the case r * < 1.

From the point of view of the construction of σ ε we need to replace the functions ζ ε with ζε , satifying

ζε (t) = 1 for t ≤ ε or t ≥ L -ε, ζε (t) = 0 for 2 ε ≤ t ≤ L -2 ε, and ζ ε ≤ 1 ε .
This choice ensures that σ ε has all the properties previously obtained with r * ε replaced by ε accordingly. Define

w ε (t) :=    η, for t ≤ √ 3ε, 1 -η r - √ 3 (t - √ 3) + η, for √ 3ε ≤ t ≤ r,
and set

ϕ ε = min{ϕ ε (d(x, I)), w ε (|x|), w ε (|x -(L, 0)|)}.
With these choices for ϕ ε and σ ε the estimates follow analogously with small differences in the constants.

Γ-limsup inequality

O L ε 2ε L -2ε L -ε Ir * ε Ir B √ 3ε (0; 0) B √ 3ε (L; 0) O L ε 2ε L -2ε L -ε Ir * ε Ir Dε D ε Ω Figure 2
.3: On the left the striped region corresponds to supp(σ ε ), remark that the balls of radius √ 3ε centered respectively in (0; 0) and (L; 0) contain the modifications we have performed to satisfy the constraint. On the right we illustrate the level-lines of the cutoff function ζε in grayscale.

Step 2. (Case of a generic σ in polyhedral form.) Indeed, in force of the results quoted in Chapter 1 it is sufficient to show equation (2.3) for a polyhedral vector measure. Following the same notation introduced therein let

σ = N j=1 m j H 1 Σ j τ j .
With no loss of generality we can assume that the segments Σ j intersect at most at their extremities. We consider measures σ satisfying constraint (2.4) so that if a point P belongs to Σ j 1 , . . . , Σ j P it must satisfy of Kirchhoff law,

j P j 1 z j m j = c i , if P ∈ S . 0, otherwise. (2.29)
where z j , is +1 if P is the ending point of the segment Σ j with respect to its orientation, and -1 if it is the starting point. Let σ j ε and ϕ j ε be the sequences constructed above for each segment Σ k and define

σ ε = N j=1 σ j ε and ϕ ε = min j ϕ j ε .
Let P j and Q j be respectively the initial and final point of the segment Σ j and recall that, by the construction made above, for each j

∇ • σ j ε = m j δ P j -δ Q j * ρ ε then by linearity of the divergence operator, it holds ∇ • σ ε = N j=1 ∇ • σ j ε = N j=1 m j δ P j -δ Q j * ρ ε
and the latter satisfies constraint (2.1) in force of equation (2.29). To conclude let us prove that lim sup

ε↓0 F ε,β (σ ε , ϕ ε ; Ω) ≤ N j=1 h β (m j )H 1 (Σ j ).
(2.30)

Chapter 2. Multidimensional case Indeed the following inequality holds true

F ε,β (σ ε , ϕ ε ; Ω) ≤ N j=1 F ε,β (σ ε , ϕ j ε ; Ω). Suppose supp(σ j 1 ε ) ∩ supp(σ j 2 ε ) ∩ • • • ∩ supp σ j P ε =
∅ for some j 1 , . . . , j P and all ε. Let r j 1 * , . . . , r j P * be the radii introduced above for each of these measures, let r * = max{r j 1 * , . . . , r j P * , 1} , set m = max{m j 1 , . . . , m j P } and consider D j 1 , . . . , D j P as defined previously. Since

j P k=1 σ k ε 2 ≤ C j P k=1 σ k ε 2
and ϕ ε ≤ ϕ j ε for any j, we have the following inequality

F ε,β (σ ε , ϕ ε ; supp(σ j 1 ε ) ∩ • • • ∩ supp(σ j P ε )) ≤ C j P k=j 1 F ε,β (σ k ε , ϕ k ε ; D k ).
And by inequality (2.28) follows

F ε,β (σ ε , ϕ ε ; supp(σ j 1 ε ) ∩ • • • ∩ supp(σ j P ε )) ≤ C (1 -η) 2 ε n-1 j P k=j 1 L n (D k ) + 2 m 2 r 2 * η ε n-2 ,
which vanishes as ε ↓ 0. Let us remark that the intersection supp(σ

j 1 ε ) ∩ supp(σ j 2 ε ) ∩ • • • ∩ supp σ j P
ε is non empty for any ε only if the segments Σ j 1 , . . . , Σ j P have a common point. Since we are considering a polyhedral vector measure composed by N segments the worst case scenario is that we have 2N intersections in which at most N segments intersects. We conclude

F ε,β (σ ε , ϕ ε ; Ω) ≤ N j=1 F ε,β (σ j ε , u j ε ; Ω) + C(N ) (1 -η) 2 ε n-1 j P k=j 1 L n (D k ) + 2 m 2 r 2 * η ε n-2
which, passing to the limit, yields inequality (2.30).

Chapter 3

The k-dimensional problem

Introduction

In this chapter we analyze how to address the problem of approximating the k-dimensional Plateau problem. In particular we aim at extending Theorems 2.1, 2.2 and 2.3 in the case where the 1-currents (vector measures) are replaced with k-currents. Let σ 0 ∈ P k (Ω) a polyhedral k-current with finite mass and let S := supp(∂σ 0 ) be compactly contained in Ω. We want to minimize a functional of the type (10) where the set of candidates ranges among all currents D k (Ω) such that

∂σ = ∂σ 0 in D k (R n ). (3.1)
Let us introduce a parameter η = η(ε) which satisfies

η(ε) = βε n-k+1 for β ∈ R + (3.2)
and let X ε (Ω) be the set of pairs (σ ε , ϕ ε ) where ϕ ε ∈ W 1,p (Ω, [η, 1]) and has trace 1 on ∂Ω and σ ε is of finite mass with density absolutely continuous with respect to L n . In this case we identify the current σ ε with its L 1 (Ω, Λ k (R n )) density. Furthermore as in equation (2.1) given a convolution kernel ρ ε we impose the constraint

∂σ ε = (∂σ 0 ) * ρ ε in D k (R n ). For (σ ε , ϕ ε ) ∈ D k (Ω) × L 2 (Ω) let F k ε,β (σ ε , ϕ ε ; Ω) :=    Ω ε p-n+k |∇ϕ ε | p + (1 -ϕ ε ) 2 ε n-k + ϕ ε |σ ε | 2 ε dx, if (σ ε , ϕ ε ) ∈ X ε (Ω),
+∞, otherwise.

(3.3) Let us denote with X the set of pairs (σ, ϕ) such that σ is a k-rectifiable current satisfying (3.1) and ϕ ≡ 1. In this section we show that for any sequence ε ↓ 0 the Γ-limit of the family (F k ε,β ) ε∈R + is the functional

E k β (σ, ϕ; Ω) =    supp σ h n-k β (m(x)) dH k (x), if (σ, ϕ) ∈ X, +∞, otherwise in M(Ω, R n ) × L 2 (Ω), (3.4) Chapter 3. The k-dimensional problem
where the function h n-k β : R + → R + is the function obtained in Appendix B for the choice d = nk and is endowed with the same properties stated in Chapter 2. In particular under the assumption p > nk we first prove a compactness theorem.

Theorem 3.1. Assume that β > 0. For any sequence ε ↓ 0, (σ ε , ϕ ε ) ∈ D k (Ω) × L 2 (Ω) such that F k ε,β (σ ε , ϕ ε ; Ω) ≤ F 0 < +∞ then ϕ ε → 1 in L 2 (Ω)
and there exists a rectifiable k-current σ ∈ D k (Ω) such that, up to a subsequence, σ ε * σ and (σ, 1) ∈ X.

Then we show the Γ-convergence result, namely Theorem 3.2. Assume that β ≥ 0.

1. For any (σ, ϕ) ∈ D k (Ω) × L 2 (Ω) and any sequence (σ ε , ϕ ε ) ∈ D k (Ω) × L 2 (Ω) such that (σ ε , ϕ ε ) → (σ, ϕ) it holds lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; Ω) ≥ E k β (σ, ϕ; Ω).

For any couple

(σ, ϕ) ∈ D k (Ω) × L 2 (Ω) there exists a sequence (σ ε , ϕ ε ) ∈ D k (Ω) × L 2 (Ω) such that (σ ε , ϕ ε ) → (σ, ϕ) and lim sup ε↓0 F k ε,β (σ ε , ϕ ε ; Ω) ≤ E k β (σ, ϕ; Ω).

Compactness and k-rectifiability

Proof of Proposition 3.1. By the same procedure of Lemma 2.1 we obtain

|σ ε |(Ω) ≤ F 0 2 + F 0 2 a (1 -λ) 2 + |Ω| ε F 0 λ (3.5)
and

Ω (1 -ϕ ε ) 2 ≤ ε n-k F 0 .
Therefore by the weak compactness of D k (Ω) we obtain the existence of a limit kcurrent σ a limit measure µ and a subsequence ε such that

σ ε * σ, |σ ε | * µ.
As in the 1-dimensional case it is still necessary to prove the rectifiability of the limit current. This is obtained by showing that the support of σ is of finite size.

Step 1. (Preliminaries and good representative for v ∈ Λ k (R n ).) Let us introduce the set

I := {I = (i 1 , . . . , i k ) : 1 ≤ i 1 < i 2 < • • • < i k ≤ n}.
and denote

e I = e i 1 ∧ • • • ∧ e i k . So that Λ k (R n ) is the Euclidean space with basis {e I } I∈I . Let v ∈ Λ k (R n
) and consider the problem

a 0 = max{a ∈ R : v = af 1 ∧ • • • ∧ f k + t : (f 1 , . . . , f n ) orthonormal basis, t ∈ (f 1 ∧ • • • ∧ f k ) ⊥ }.

Compactness and k-rectifiability

Notice that a 0 ≥ 1/ |I|. Assume that the optimum for the preceding problem is obtained with (f 1 , . . . , f n ) = (e 1 , . . . , e n ). We note

v = a 0 e I 0 + i∈I 1 a I e I + I∈J
a I e I with I 0 = (1, . . . , k) and

I 1 := {I = (i 1 , . . . , i k ) ∈ I : 1 ≤ i 1 < • • • < i k-1 ≤ k < i k ≤ n}, J := I \ (I 1 ∪ {I 0 }).
We claim that a I = 0 for I ∈ I 1 . Indeed, let I 1 = (e 1 , . . . , e l-1 , e l+1 , . . . , e k , e h ) ∈ I 1 and for φ ∈ R, let e φ be orthonormal base defined as

e i = e φ i for i = {l, h}, e l = cos(φ)e φ l -sin(φ)e φ h , e h = sin(φ)e φ l + cos(φ)e φ h .
In this basis

v = a 0 cos(φ) + a I 1 (-1) k-l sin(φ) e φ I 0 + t φ , with w φ ∈ (e φ ) ⊥ .
By optimality of (e 1 , . . . , e n ) we deduce a I 1 = 0 which proves the claim. Hence we write v = a 0 e I 0 + t, with t ∈ span{e I : I ∈ J }.

(3.6)

Now we let Let θ ∈ (0, 1/4 n ) and Σ be the set of points for which there exists a sequence r j ↓ 0 such that

σ(B r j (x)) |σ|(B r j (x)) -→ w(x) ∈ SΛ k (R n ) and |σ|(B r j /4 (x)) |σ|(B r j (x)) ≥ θ.
In particular w is a |σ|-measurable map and we have σ = w |σ| Σ.

Step 2. (Flux of σ ε trough a small (nk)-disk.) Consider a point x ∈ Σ \ S , with no loss of generality we assume x = 0. Let v = w(0), up to a change of basis, by equation (3.6) we write v = a 0 e I 0 + t, with t ∈ span{e I : I ∈ J }.

Let j sufficiently small, such that B r j ∩ S = ∅ and

σ(B r j ) • v ≥ (1 -ξ)|σ|(B r j ).
(3.7)

Set, to simplify notation, r j = r and r * = r/ √ 2. For x ∈ R n we write (x , x ) ∈ R k × R n-k for the usual decomposition and denote B r , B r the k-dimensional and the (nk)-dimensional ball respectively. Let χ ∈ C ∞ (B 1 ) be a radial cut-off function with χ(x ) = 1 for |x | ≤ 1/2 and χ(x ) = 0 for |x | ≥ 3/4. Set χ r * (x ) = χ(x /r * ), then since σ ε is a L 1 function for ε > 0 we can define

g ε (x ) := B r * χ r * (x ) σ ε , e I 0 dx = B r * χ r * (x )σ 0 ε dx (3.8)
for any x ∈ B r * . Let us compute ∂ l g ε (x ) for l ∈ {1, . . . , k}. Since ∂σ ε = 0 in B r , it holds σ ε , dω = 0 for any smooth (k -1)-differential form ω ∈ D k-1 (B r ). Choosing ω of the form

ω = β(x) dx 1 ∧ . . . dx l-1 ∧ dx l+1 ∧ • • • ∧ dx k (3.9) we obtain dω = (-1) l-1 ∂ l β(x) dx 1 ∧ • • • ∧ dx k + + (-1) k-1 d h=k+1 ∂ h β(x) dx 1 ∧ . . . dx l-1 ∧ dx l+1 ∧ • • • ∧ dx k ∧ dx h . Denote σ I ε = σ, e I , then imposing σ ε , dω = 0 for every β ∈ C ∞ c (B r ) in (3.9) yields (-1) k-l ∂ l σ 0 ε + h∈{k+1,...,d} I=(1,...,l-1,l+1,...,k,h) ∂ h σ I ε = 0.
Hence,

∂ l g ε (x ) = (-1) k-l r * h∈{k+1,...,d} I=(1,...,l-1,l+1,...,k,h) B r * ∂ h χ r * (x )σ I ε dx . (3.10)
Let us introduce the notation σ I 1 ε :=

I∈I 1 σ I ε e I ,
denoting with ∇ the gradient with respect to x , equation (3.10) rewrites as

∇ g ε (x ) = 1 r * B r * Y x r * σ I 1 ε dx . (3.11)
Where Y is smooth and compactly supported in B 1 and with values into the linear maps : span{e I : I ∈ I 1 } → R k . Let us prove that, for some r, the functions g ε converge in BV- * to some g. First for a.e. choice of r ∈ [(1ξ)r * , r * ] it must hold µ(∂B r * × B r * ) = 0 so that

g ε (x ) = B r * χ r * (x ) σ ε , e I 0 dx ε↓0 --→ B r * χ r * (x ) d σ, e I 0 =: g(x ).
(3.12)

Secondly we define the mean value

g := 1 |B r| B r g(x ) dx = 1 |B r| B r B r * χ r * (x ) dσ 0 dx .
and taking advantage of (3.7) and the definition of Σ, we see that

g ≥ θ |I| -ξ |σ|(B r ) |B r| > 0.
On the other hand, denoting Π : R n → R n-k , x → x , from (3.6), we have

|Πσ|(B r × B r * ) ≤ 3ξ θ |I| -ξ |B r| g.
Now from (3.11) --(3.12) and the latter we obtain

D g, φ = 1 r * B r ×B r * φ(x ) Y x r * dσ I 1 and |D g|(B r) ≤ C |B r| √ ξ g r * .
Finally from Poincaré -Wirtinger inequality and the convergence g ε → g in L 1 (B r) is easy to show that for any sufficiently small ε the sets

A ε = x ∈ B r : g ε (x) ≥ g 8 are such that |A ε | ≥ |B r|/2.
Step 3. (Conclusion.) Set ϑ ε (x , x ) = |χ r * (x )σ 0 ε | and observe that for fixed x by construction

Br * ϑ ε (x , x ) dx = g ε (x ). Therefore for any x ∈ A ε it holds Br * ϑ ε (x , x ) dx ≥ g/8. Furthermore supp(ϑ ε (x )) ⊂ B r with r := 3 4 r * < r * . Now, by Fubini F k ε,β (σ ε , ϕ ε ; B r ) ≥ Aε B r ε p-n+k |∇ϕ ε | p + (1 -ϕ ε ) 2 ε n-k + ϕ ε |σ ε | 2 ε dx dx ≥ Aε B r * ε p-n+k |∇ϕ ε | p + (1 -ϕ ε ) 2 ε n-k + ϕ ε |ϑ ε (x , x )| 2 ε dx dx
(3.13) With the notation introduced in section 2.2 and by definition of

A ε F ε,β (σ ε , ϕ ε ; B r ) ≥ Aε inf (ϑ,ϕ)∈Y ε,β (g/8,r,r) G k ε,β (ϑ, ϕ) dx = Aε h n-k ε,β (g/8, r, r) dx = h n-k ε,β (g/8, r, r) |A ε |.
Taking the infimum limit, by Proposition 2.1, in particular equation (2.12) we get

lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; B r ) ≥ lim inf ε↓0 h n-k ε,β (g/8, r, r) |A ε | ≥ κ |B r| 2 . (3.14)
Recall that the latter stands for a.e. r ∈ [(1ξ)r * , r * ] and r * = r/ √ 2 thus we may rewrite

lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; B r ) ≥ κ ω k r k 2 1+k/2
. As in Lemma 2.3 we conclude applying Besicovitch theorem to obtain H k (Σ) < +∞. Finally, thanks to the latter and equation (3.5), White's rectifiability theorem [Whi99b, Thm 6.1] applies and σ is a k-rectifiable current.

Γ-liminf inequality

Proof of item 1) of Theorem 3.2. With no loss of generality we assume that lim inf ε↓0 F k ε,β (σ ε , ϕ ε ) < +∞ otherwise the inequality is trivial. For a Borel set A ⊂ Ω, we define

H k (A) := lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; A),
so that H k is a subadditive set function. By assumption, the limit current σ is krectifiable; we write σ = m τ H k Σ. We claim that lim inf r↓0

H k B(x, r) ω k r k ≥ h n-k β (m(x)) for H k -almost every x ∈ Σ. (3.15)
Assuming the latter the proof is achieved as in Theorem 2.2. To establish the claim (3.15) we restrict our attention to a single point and we assume x = 0, m = m(0) and τ (0) = e 1 ∧ • • • ∧ e k then for any ξ > 0 there exists r 0 = r(ξ) such that

σ, e 1 ∧ • • • ∧ e k (B r ) ≥ (1 -ξ)|σ|(B r ) and (1 -ξ) m ≤ |σ|(B r ) ω k r k ≤ (1 + ξ) m, for r ≤ r 0 . (3.16)
Let δ be an infinitesimal quantity and set, for r < r 0 , r = √ 1δ 2 r and r = δr and define the cylinder

C δ,r (e 1 , ∧ • • • ∧ e n ) = C δ,r := (x ; x ) ∈ R k × R n-k : |x | ≤
∇ g ε (x ) = 1 r B r Y x r dσ I 1 ε .
For a.e. choice of δ it holds |σ|(∂B r × B r ) = 0 therefore, for any such choice, γ ε converges in BV (B r) to

g(x ) := B r χ r(x ) dσ 0 and D g, φ = 1 r B r ×B r φ(x ) Y x r dσ I 1 .
Now we use (3.16) to improve the estimates on g and |D g|. Indeed, for δ sufficiently small, r < r/2 therefore B r ⊂ B r × B r and

lim ε↓0 g ε ≥ (1 -ξ) 1 |B r | B r ×B r χ r * (x ) d|σ| ≥ (1 -ξ) 2 m.
3.4. Γ-limsup inequality and denoting Π :

R n → R n-k , x → x we have |Πσ|(C r ) ≤ (1 + ξ) 3ξ |B r| m and |D g|(B r) ≤ C |B r| √ ξ m r .
Choose r sufficiently small then by Poincaré -Wirtinger inequality there exists a set A of almost full measure in B r such that g ε (x ) ≥ (1ξ) 2 m, and following the proof of the previous lemma (Step 3) up to equation (3.14) we get

lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; B r ) ≥ lim inf ε↓0 h n-k ε,a (1 -ξ) 2 m, r, r |A|.
Since ξ and δ are arbitrary and |A| can be chosen arbitrary close to |B r| applying Proposition 2.1 with d = nk to the latter we conclude

lim inf ε↓0 F k ε,β (σ ε , ϕ ε ; B r ) ≥ h n-k β (m) ω k r k .

Γ-limsup inequality

For the lim-sup inequality, we start by approximating σ with a polyhedral current: given δ > 0, there exists a k polyhedral current σ satisfying ∂ σ = ∂σ 0 and with F(σσ) < δ and E a (σ) < E a (σ) + ε. This result of independent interest is established in [START_REF] Chambolle | Strong approximation in h-mass of rectifiable currents under homological constraint[END_REF]. A similar result has been proved recently by Colombo et al. in [CDRMS17, Prop. 2.6] (see also [START_REF] White | Rectifiability of flat chains[END_REF] Section 6]). The authors build an approximation of a k-rectifiable current in flat norm and in energy but their construction creates new boundaries and can not ensure the condition ∂σ = ∂σ 0 .

Proof of item 2) of Theorem 3.2: By [CFM18, Theorem 1.1 and Remark 1.6] we can assume that σ is a polyhedral current. We show how to produce the approximating (σ ε , ϕ ε ) for σ supported on a single k-dimensional simplex Q. We assume with no loss of generality that Q ⊂ R k , and that σ writes as

m H k Q ∧ (e 1 ∧ • • • ∧ e k ).
For δ > 0 fixed, we consider the optimal profiles

ϕ ε (t) :=          η, for 0 ≤ t ≤ r * ε, v δ t ε , for r * ε ≤ t ≤ r, 1 for r ≤ t, and
ϑ ε = m χ B r * ε (x ) ω n-k (εr * ) n-k
with r * and v δ , defined in Proposition 2.2 for the choice d = nk. We denote ∂Q the relative boundary of Q and given a set S we write d(x, S) for the distance function from S. Recall that we use the notation S t for the t-enlargement of the set S and S to denote its projection into R k . We first assume, as did for the case k = 1, r * ≥ 1, and introduce ζ ε a 0-form depending on the first k variables x , satisfying

ζ ε (x ) = 1, for x ∈ (∂Q) r * ε := {x ∈ Ω : d(x , ∂Q) ≤ r * ε} , ζ ε (x ) = 0, for x ∈ Ω \ (∂Q) 2r * ε , | dζ ε | ≤ 1 r * ε .
Then we proceed by steps, first set σ 1 ε := (|σ| * ρ ε )

σ 1 ε = σ 1 ε e 1 ∧ • • • ∧ e k and σ 2 ε (x , x ) = ϑ ε (|x |) ∧ (e 1 ∧ • • • ∧ e k ).
and observe that supp(σ

1 ε ) ∪ supp(σ 2 ε ) ⊂ Q r * ε , both σ 1
ε and σ 2 ε are radial in x and with a small abuse of notation we denote

σ 1 ε (x , s) = σ 1 ε (x , |x |), finally for any x {x }×B r * ε [σ 1 ε (x , |x |) -ϑ ε (|x |)] dx = 0.
Now we take advantage of ζ ε in order to interpolate between σ 1 ε and σ 2 ε , note that such interpolation may affect the boundary of the new current therefore we first introduce σ 3 ε which corrects this defect. In particular set

σ 3 ε (x , x ) = - n i=k+1 x i |x | n-k |x | 0 s n-k-1 σ 1 ε (x , s)ϑ ε (s) dζ ε ds ∧ e i , and
σ ε = σ 1 ε ζ ε + σ 2 ε (1 -ζ ε ) + σ 3 ε .
With this choice by a calculation similar to equation (2.27) it holds

∂σ ε = -∂σ * ρ ε ζ ε -σ 1 ε dζ ε -∂σ 2 ε (1 -ζ ε ) =0 +σ 2 ε dζ ε + ∂σ 3 ε = (∂σ) * ρ ε .
On the other hand the phase-field is simply defined as ϕ ε (x) = ϕ ε (d(x, Q)). In the case r * < 1 we need to modify the construction. For σ ε it is sufficient to replace every occurrence of ζ ε with ζε , which satisfies

ζε (x ) = 1, for x ∈ (∂Q) ε := {x ∈ Ω : d(x , ∂Q) ≤ ε} , ζε (x ) = 0, for x ∈ Ω \ (∂Q) 2ε , d ζε ≤ 1 ε . Now let w ε (t) :=    η, for t ≤ √ 3ε, 1 -η r - √ 3 (t - √ 3) + η, for √ 3ε ≤ t ≤ r.
and set

ϕ ε = min{ϕ ε (d(x, Q)), w ε (d(x, ∂Q))}.

Discussion about the results

Remark 1. Given a polyhedral current σ such that ∂σ = ∂σ 0 we perform our construction on each simplex and define σ ε as the sum of these elements. The linearity of the boundary operator grants that ∂σ ε = ∂σ 0 * ρ ε . The phase field is chosen as the pointwise minimum of the local phase fields. Finally the estimation for the Γ-limsup inequality is achieved in the same manner as Theorem 2.3.

Discussion about the results

By Lemma B.4 for any fixed d = nk the cost function h d β pointwise converges as β ↓ 0 to the function

h(m) = κ, for m > 0, 0, if m = 0,
where κ is the constant value obtained in Proposition 2.1 and depends on d. This condition is sufficient to prove that the family of functionals E k β , parametrized in β, Γ-converges to the functional

E k (σ; Ω) :=    κ H k (Σ ∩ Ω), for σ = m τ H k Σ, +∞, otherwise.
As a matter of fact for any sequence

σ β * σ in D k (Ω) it holds lim inf β↓0 E k β (σ; Ω) ≥ E k (σ; Ω) since h d β (m) ≥ κ.
On the other hand setting σ β := σ we construct a recovery sequence for any σ and obtain the Γ-limsup inequality

lim sup β↓0 E k β (σ β ; Ω) = lim sup β↓0 E k β (σ; Ω) = E k (σ; Ω).
This allows to interpret our result as an approximation of the Plateau problem in any dimension and co-dimension.

Chapter 4

Piecewise affine cost functions

Introduction

The present chapter is the result of a collaboration with Benedikt Wirth and Carolin Rossmanith from Mnster University. We generalize the approach of Chapter 1 to the case in which the cost function h is piecewise affine. Let N ∈ N and

∞ ≥ α 0 > α 1 > • • • > α N > 0, 0 = β 0 < β 1 < • • • < β N < ∞, we define the piecewise affine transport cost h : [0, ∞) → [0, ∞), h(m) = min i=0,...,N {α i m + β i } . If α 0 = ∞ we interpret h as h(m) = 0 if m = 0 , min i=1,...,N {α i m + β i } else.
We first remark that if α 0 = ∞ the right derivative of h in the origin diverges, then E h (σ) is finite if and only if σ is a rectifiable vector measure as stated in [CDRMS17, Proposition 2.7]. On the contrary, in the case α 0 < ∞, the energy E h may be finite on more complicated structures. Consider the usual probability measures µ + and µ - which, in the case α 0 < ∞ may be considered diffuse, and let σ be a vector measure satisfying the constraint ∇ • σ = µ +µ -. Recall that in force of the Generalized Gilbert-Steiner formula presented in Proposition 0.1 and our choice of h we have

E h (σ) = α 0 |σ ⊥ |(Ω) + Σ h(m) dH 1 ,
in the latter α 0 = h (0), the right derivative in 0. Above we have decomposed σ as

σ = σ ⊥ + mτ H 1 Σ where (m, τ, Σ) is an H 1 rectifiable measure and σ ⊥ is H 1 -diffuse.
The functional proposed in this chapter resembles the one introduced in the first chapter. Let us remark that in the first chapter we were able to recover in the limit any affine cost function of the form 1 + β|m|. Here we rescale the functional presented therein and introduce multiple phase fields, each one responsible for a different component of the piecewise affine cost function. The constraint component is modified to Chapter 4. Piecewise affine cost functions let interact the different phase fields. Let us be more formal. We let X µ + ,µ - ε denote the space of (N + 1)-uples (σ, ϕ 1 , . . . , ϕ N ) where σ ∈ M(Ω, R 2 ) such that

∇ • σ = (µ + -µ -) * ρ ε and ϕ i ∈ L 1 (Ω) for each i = {1, . . . , N }. Eventually we let F ε (σ, ϕ 1 , . . . , ϕ N ) = Ω ω ε α 0 , γ ε (x) ε , |σ(x)| dx + N i=1 β i T ε (ϕ i ) ,
where we abbreviated (with some p > 1)

T ε [ϕ] = 1 2 Ω ε|∇ϕ(x)| 2 + (ϕ(x) -1) 2 ε dx , γ ε (x) = min i=1,...,N ϕ i (x) 2 + α 2 i ε 2 /β i , ω ε α 0 , γ ε (x) ε , |σ(x)| = γε ε |σ| 2 2 if |σ| ≤ α 0 γε/ε α 0 (|σ| -α 0 2γε/ε ) if |σ| > α 0 γε/ε + ε p |σ(x)| 2 for α 0 < ∞ , ω ε α 0 , γ ε (x) ε , |σ(x)| = γ ε ε |σ| 2 2 for α 0 = ∞ . (4.1)
Remark 2 (Motivation of ω ε α 0 , γε(x) ε , |σ(x)| via relaxation). Keeping the phase fields ϕ 1 , . . . , ϕ N fixed and ignoring the regularizing term ε p |σ| 2 , the integrand ω ε (α 0 , γε ε , |σ|) is the convexification in σ of min α 0 |σ|,

(ϕ 2 1 + α 2 1 ε 2 /β 1 )|σ| 2 2ε , . . . , (ϕ 2 N + α 2 N ε 2 /β N )|σ| 2 2ε = min α 0 |σ|, γ ε ε |σ| 2 2 ,
which shows the intuition of the phase field functional much clearer. Indeed, the minimum over N + 1 terms parallels the minimum in the definition of h, and the i th term for i = 0, . . . , N describes (part of) the transportation cost α i m + β i . However, since the above is not convex with respect to σ, a functional with this integrand would not be weakly lower semi-continuous in σ and consequently possess no minimizers in general. Taking the lower semi-continuous envelope corresponds to replacing the above by ω ε (α 0 , γε ε , |σ|) (note that this only ensures existence of minimizers, but will not change the Γ-limit of the phase field functional).

Note that the pointwise minimum inside γ ε is well-defined almost everywhere, since all elements of X µ + ,µ - ε are Lebesgue-measurable. Note also that for fixed phase fields ϕ 1 , . . . , ϕ N the phase field cost functional F ε is convex in σ. This ensures the existence of minimizers for F ε , which follows by a standard application of the direct method as it will be shown in the following section. Eventually we extend the functional on the whole M(Ω, R 2 ) × L 1 (Ω) N letting (N + 1)-uples (σ, ϕ 1 , . . . , ϕ N ) we let

F ε (σ, ϕ 1 , . . . , ϕ N ) = ∞ 4.2. Remarks if (σ, ϕ 1 , . . . , ϕ N ) ∈ X ε .
For consistency we introduce the set X consisting of those (N + 1)-tuples (σ, ϕ 1 , . . . , ϕ N ) such that each ϕ i = 1 for each i and σ ∈ M(Ω, R n ) is a vector valued measure satifing the constraint

∇ • σ = µ + -µ -.
Then we define the limit energy

E (σ, ϕ 1 , . . . , ϕ N ) := E h (σ), if (σ, ϕ 1 , . . . , ϕ N ) ∈ X, +∞, otherwise. (4.2)
In particular in this chapter we will prove the following theorems.

Theorem 4.1 (Convergence of phase field cost functional). For admissible µ + , µ -∈ P(Ω) we have

Γ -lim ε→0 F ε = E ,
where the Γ-limit is with respect to weak- * convergence in M(Ω; R 2 ) and strong convergence in L 1 (Ω) N .

In the above we say admissible since we will implicitly assume that if α 0 = ∞ then µ -and µ + are supported on a countable number of points. The proof of this result is provided in the next section. Together with the following equicoercivity statement, whose proof is also deferred to the next section, we have that minimizers of the phase field cost functional F ε approximate minimizers of the original cost functional E . Theorem 4.2 (Equicoercivity). For ε → 0 let (σ ε , ϕ ε 1 , . . . , ϕ ε N ) be a sequence with uniformly bounded phase field cost functional F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ) < C < ∞. Then, along a subsequence, σ ε * σ in M(Ω; R 2 ) for some σ ∈ M(Ω; R 2 ) and ϕ ε i → 1 in L 1 (Ω), i = 1, . . . , N . As a consequence, if µ + , µ -∈ P(Ω) are admissible and such that there exists σ ∈ M(Ω, R 2 ) with E (σ, 1, . . . , 1) < ∞, then any sequence of minimizers of F ε contains a subsequence converging to a minimizer of E as ε → 0.

Remarks

Before moving to the proofs of the results previously stated let us stress out some important remarks.

Proposition 4.1 (Existence of minimizers to the phase field functional). The phase field cost functional F ε has a minimizer (σ, ϕ 1 , . . . , ϕ N ) ∈ X ε .

Proof. The functional is bounded below by 0 and has a nonempty domain. Indeed, choose φ1 ≡ . . . ≡ φN ≡ 1 and σ = ∇ψ, where ψ solves ∆ψ = µ ε +µ ε -in Ω with Neumann boundary conditions ∇ψ • ν ∂Ω = 0, ν ∂Ω being the unit outward normal to ∂Ω. (Since Ω µ ε +µ ε -dx = 0, a solution ψ exists and lies in W 2,2 (Ω) by standard elliptic regularity.) Obviously, (σ, φ1 , . . . , φN )

∈ X µ + ,µ - ε with F ε (σ, φ1 , . . . , φN ) < ∞. Now consider a minimizing sequence (σ k , ϕ k 1 , . . . , ϕ k N ) ∈ X µ + ,µ - ε , k ∈ N, with F ε (σ k , ϕ k 1 , . . . , ϕ k N ) → inf F ε monotonically as k → ∞. Since F ε is coercive with respect to H = L 2 (Ω; R 2 ) × W 1,2 (Ω) N , (σ k , ϕ k 1 , . . . , ϕ k N
) is uniformly bounded in H so that we can extract a weakly converging subsequence, still indexed by k for simplicity,

(σ k , ϕ k 1 , . . . , ϕ k N ) (σ, ϕ 1 , . . . , ϕ N ) .
Due to the closedness of X µ + ,µ - ε with respect to weak convergence in H we have (σ, ϕ 1 , . . . , ϕ N ) ∈ X µ + ,µ - ε . Note that the integrand of F ε is convex in σ(x) and the ∇ϕ i (x) as well as continuous in σ(x) and the ϕ i (x), thus F ε is lower semi-continuous along the sequence. Indeed, consider a subsequence along which each term T ε [ϕ k i ] converges and along which the ϕ k i converge pointwise almost everywhere (so that also

γ k ε (x) = min i=1,...,N ϕ k i (x) 2 + α 2 i ε 2 /β i converges for almost all x ∈ Ω)
. By Mazur's lemma, a sequence of convex combinations m k j=k λ k j σ j of the σ k converges strongly (and up to another subsequence again pointwise) so that by Fatou's lemma we have

inf F ε = lim k→∞ F ε (σ k , ϕ k 1 , . . . , ϕ k N ) = lim k→∞ Ω ω ε α 0 , γ k ε (x) ε , |σ k (x)| dx + N i=1 β i lim k→∞ T ε [ϕ k i ] ≥ lim k→∞ m k j=k λ k j Ω ω ε α 0 , γ j ε (x) ε , |σ j (x)| dx + N i=1 β i T ε (ϕ i ) ≥ Ω lim inf k→∞ m k j=k λ k j ω ε α 0 , γ j ε (x) ε , |σ j (x)| dx + N i=1 β i T ε (ϕ i ) ≥ Ω lim inf k→∞ m k j=k λ k j ω ε α 0 , inf i=k,...,m k γ i ε (x) ε , |σ j (x)| dx + N i=1 β i T ε (ϕ i ) ≥ Ω lim inf k→∞ ω ε α 0 , inf i=k,...,m k γ i ε (x) ε , m k j=k λ k j |σ j (x)| dx + N i=1 β i T ε (ϕ i ) = F ε (σ, ϕ 1 , . . . , ϕ N ) ,
where we exploited the weak lower semi-continuity of T ε , the monotonicity of ω ε α 0 , γε(x) ε , |σ(x)| in its second argument, its convexity in its last argument, and its continuity in its latter two arguments.

Remark 3 (Regularization of σ). Note that the phase field cost functional F ε is L 2 (Ω; R 2 )-coercive in σ, which is essential to have sequentially weak compactness of subsets of X µ + ,µ - ε with finite cost (and as a consequence existence of minimizers). For α 0 < ∞ this is ensured by the regularization term ε p |σ| 2 (which has no other purpose). Without it, the functional would only feature weak- * coercivity for σ in M(Ω; R 2 ), however, the integral Ω ω ε α 0 , γε(x) ε , |σ(x)| dx with γ ε Lebesgue-measurable would in general not be well-defined for σ ∈ M(Ω; R 2 ).

Remark 4 (Phase field boundary conditions). Recall that we imposed boundary conditions ϕ i = 1 on ∂Ω. Without those, the recovery sequence from the following section could easily be adapted such that all full phase field profiles near the boundary will be 4.3. The Γ-limit of the phase field functional replaced by half, one-sided phase field profiles. It is straightforward to show that the resulting limit functional would become

Σ∩Ω min{α 0 m, α 1 m + β 1 , . . . , α N m + β N } + Σ∩∂Ω min{α 0 m, α 1 m + β 1 /2, . . . , α N m + β N /2} + α 0 |σ ⊥ |(Ω) ,
where fluxes along the boundary are cheaper and thus preferred.

Remark 5 (Divergence measure vector fields and flat chains). Any divergence measure vector field can be identified with a flat 1-chain or a locally normal 1-current (see for instance [ Š07, Sec. 5] or [BW17, Rem. 2.29(3)]; comprehensive references for flat chains and currents are [START_REF] Whitney | Geometric integration theory[END_REF][START_REF] Federer | Geometric measure theory[END_REF]). Furthermore, for a sequence σ j , j ∈ N, of divergence measure vector fields with uniformly bounded ∇•σ j M , weak- * convergence is equivalent to convergence of the corresponding flat 1-chains with respect to the flat norm [BW17, Rem. 2.29(4)]. Analogously, scalar Radon measures of finite total variation and bounded support can be identified with flat 0-chains or locally normal 0-currents [Whi99b, Thm. 2.2], and for a bounded sequence of compactly supported scalar measures, weak- * convergence is equivalent to convergence with respect to the flat norm of the corresponding flat 0-chains.

From the above it follows that in Theorems 4.1 and 4.2 we may replace weak- * convergence by convergence with respect to the flat norm. Indeed, for both results it suffices to consider sequences (σ ε , ϕ ε 1 , . . . , ϕ ε N ) with uniformly bounded cost F ε . For those we have uniformly bounded σ ε M (by Theorem 4.2) as well as uniformly bounded ∇ • σ ε M = µ ε +µ ε -M so that weak- * and flat norm convergence are equivalent.

The Γ-limit of the phase field functional

In this section we prove the Γ-convergence result. As it is canonical, we begin with the lim inf-inequality, after which we prove the lim sup-inequality as well as equicoercivity.

The Γ-lim inf inequality for the dimension-reduced problem

Here we consider the energy reduced to codimension-1 slices of the domain Ω. In our particular case of a two-dimensional domain, each slice is just one-dimensional, which will simplify notation a little (the procedure would be the same for higher codimensions, though). The reduced functional depends on the (scalar) normal flux ϑ through the slice as well as the scalar phase fields ϕ 1 , . . . , ϕ N restricted to the slice. First observe that any measure ϑ ∈ M(I) can be decomposed into its atoms and a remainder, namely

ϑ = m ϑ H 0 S ϑ + ϑ ⊥ ,
where S ϑ ⊂ I is the set of atoms of ϑ, m ϑ : S ϑ → R is H 0 S ϑ -measurable, and ϑ ⊥ contains no atoms. Analogously to the functional introduced above we define the Chapter 4. Piecewise affine cost functions

reduced cost functional G h (•; I) : M(I) → [0, ∞), G h (ϑ; I) = x∈S ϑ ∩I h(|m ϑ (x)|) + h (0)|ϑ ⊥ |(I)
for α 0 < ∞ and otherwise

G h (ϑ; I) = x∈S ϑ ∩I h(|m ϑ (x)|) if ϑ ⊥ = 0, ∞ else.
Its extension to

M(I) × L 1 (I) N is G(•; I) : M(I) × L 1 (I) N → [0, ∞), G (ϑ, ϕ 1 , . . . , ϕ N ; I) = G h (ϑ; I) if ϕ 1 = . . . = ϕ N = 1 almost everywhere, ∞ else.
For any (ϑ, ϕ 1 , . . . , ϕ N ) ∈ L 2 (I) × W 1,2 (I) N we define the reduced phase field functional on I as

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) = I ω ε α 0 , γε(x) ε , |ϑ(x)| dx + N i=1 β i T ε [ϕ i ; I] , T ε (ϕ; I) = 1 2 I ε|ϕ (x)| 2 + (ϕ(x) -1) 2 ε dx ,
with ω ε and γ ε from equation (4.1). Eventually we extend the above functional on M(I) × L 1 (I) N by setting G ε (ϑ, ϕ 1 , . . . , ϕ N ;

I) = +∞ if (ϑ, ϕ 1 , . . . , ϕ N ) ∈ L 2 (I) × W 1,2 (I) N .
For notational convenience, we next introduce the sets K ε i on which the pointwise minimum inside G ε (or also F ε ) is realized by the i th element.

Definition 2 (Cost domains). For given (ϑ, ϕ 1 , . . . , ϕ N ) ∈ L 2 (I) × W 1,2 (I) N we set

K ε 0 = K ε 0 (ϑ, ϕ 1 , . . . , ϕ N ; I) = x ∈ I |ϑ(x)| > α 0 ε γ ε , K ε i = K ε i (ϑ, ϕ 1 , . . . , ϕ N ; I) = x ∈ I \ i-1 j=0 K ε j ϕ i (x) 2 + α 2 i ε 2 β i = γ ε (x) , i = 1, . . . , N .
The sets are analogously defined for (σ, ϕ 1 , . . . , ϕ N ) ∈ L 2 (Ω; R 2 ) × W 1,2 (Ω) N , where we use the same notation (which case is referred to will be clear from the context).

We now show the following lower bound on the energy, from which the Γlim inf inequality for the dimension-reduced situation will automatically follow. 4.3. The Γ-limit of the phase field functional

1. If α 0 < ∞ we have G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) ≥ (η -δ) 2 Iη∪C ≥ α 0 |ϑ| dx + (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx -α 2 0 H 1 (I) ε δ 2 . 2. If α 0 = ∞ we have G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) ≥ δ 2 2εH 1 (I) Iη∪C ≥ |ϑ| dx 2 + (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx -α 2 1 H 1 (I) ε δ 2 .
Proof. 1. (α 0 < ∞) We first show that without loss of generality we may assume

|ϑ| ≥ α 0 ε 2γ ε 1 1 -(η -δ) 2 on K ε 0 . (4.3)
The motivation is that there may be regions in which a phase field ϕ i is (still) small, but in which we actually have to pay α 0 |ϑ|. Thus, in those regions we would like ω ε (α 0 , γε(x) ε , |ϑ(x)|) to approximate α 0 |ϑ(x)| sufficiently well, and the above condition on ϑ ensures

ω ε α 0 , γ ε (x) ε , |ϑ(x)| = α 0 |ϑ(x)| - α 2 0 ε 2γ ε (x) + ε p |ϑ| 2 ≥ (η -δ) 2 α 0 |ϑ(x)| for x ∈ K ε 0 . (4.4)
We achieve (4.3) by modifying ϑ while keeping the cost as well as Iη |ϑ| dx and C |ϑ| dx for all C ∈ C η the same so that the overall estimate of the proposition is not affected. The modification mimics the relaxation from Remark 2: the modified ϑ oscillates between small and very large values. Indeed, for fixed

C ∈ C η ∪ {I η } and x ∈ C set θ(x) =      max{ α 0 ε 2γε(x) 1 1-(η-δ) 2 , ϑ(x)} if x ∈ K ε 0 ∩ (-∞, t C ] , α 0 ε γε(x) if x ∈ K ε 0 ∩ (t C , ∞) , ϑ (
G ε ( θ, ϕ 1 , . . . , ϕ N ; I) -G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) = K ε 0 ω ε α 0 , γ ε (x) ε , | θ(x)| -ω ε α 0 , γ ε (x) ε , |ϑ(x)| dx = K ε 0 α 0 | θ(x)| - α 2 0 ε 2γ ε (x) -α 0 |ϑ(x)| - α 2 0 ε 2γ ε (x) dx = α 0 K ε 0 | θ(x)| dx -α 0 K ε 0 |ϑ(x)| dx = 0 .
Note that the modification θ has a different set K ε 0 ( θ, ϕ 1 , . . . , ϕ N ; I) than the original ϑ. Indeed, by definition of θ we have

K ε 0 ( θ, ϕ 1 , . . . , ϕ N ; I) ⊂ K ε 0 (ϑ, ϕ 1 , . . . , ϕ N ; I) and |ϑ(x)| ≥ α 0 ε 2γε(x) 1 1-(η-δ) 2 on K ε 0 ( θ,
γ ε ≥ δ 2 on I η ∪ C ≥ we compute G ε (ϑ, ϕ 1 , . . . , ϕ N ; I η ∪ C ≥ ) ≥ (Iη∪C ≥ )∩K ε 0 ω ε α 0 , γ ε ε , |ϑ| dx + δ 2 2ε (Iη∪C ≥ )\K ε 0 |ϑ| 2 dx ≥ (η -δ) 2 Iη∪C ≥ α 0 |ϑ| dx -α 0 m 0 + δ 2 2εH 1 ((I η ∪ C ≥ ) \ K ε 0 ) m 2 0 ,
where we have employed (4.4) and Jensen's inequality. Upon minimizing in m 0 , which yields the optimal value α 0 εH 1 ((I η ∪ C ≥ ) \ K ε 0 )/δ 2 for m 0 , we thus obtain

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I η ∪ C ≥ ) ≥ (η -δ) 2 Iη∪C ≥ α 0 |ϑ| dx - α 2 0 εH 1 (I η ∪ C ≥ ) 2δ 2 .
Next consider for each C ∈ C η \ C ≥ η and i = 1, . . . , N the subsets

C ≥ i = C ∩ K ε i ∩ {ϕ i ≥ δ} , C < i = C ∩ K ε i ∩ {ϕ i <
δ} , and abbreviate m A = A |ϑ| dx for any A ⊂ I. Using Young's inequality, for i, j = 1, . . . , N we have

C ≥ i ϕ 2 i + α 2 i ε 2 /β i 2ε |ϑ| 2 dx ≥ C ≥ i δ 2 + α 2 i ε 2 /β i 2ε |ϑ| 2 dx ≥ C ≥ i α i |ϑ| - α 2 i ε/2 δ 2 + α 2 i ε 2 /β i dx ≥ α j m C ≥ i - α 2 i ε 2δ 2 H 1 (C ≥ i ) .
Similarly, using Jensen's inequality we have

C < i ϕ 2 i + α 2 i ε 2 /β i 2ε |ϑ| 2 + β i 2ε (ϕ i -1) 2 dx ≥ C < i α 2 i ε 2β i |ϑ| 2 + β i 2ε (δ -1) 2 dx ≥ α 2 i ε 2β i 1 H 1 (C < i ) C < i |ϑ| dx 2 + β i 2ε (1 -δ) 2 H 1 (C < i ) ≥ α i (1 -δ)m C < i ,
4.3. The Γ-limit of the phase field functional where in the last step we optimized for H 1 (C < i ). Finally, if inf C ϕ i ≤ δ we have (using Young's inequality)

C\C < i β i 2 ε|ϕ i | 2 + (ϕ i -1) 2 ε dx ≥ β i C\C < i |ϕ i | |1 -ϕ i | dx ≥ β i d c |ϕ i | |1 -ϕ i | dx + f e |ϕ i | |1 -ϕ i | dx ≥ β i δ η ϕ i -1 dϕ i + η δ 1 -ϕ i dϕ i = β i ((1 -δ) 2 -(1 -η) 2 ) ≥ β i (η -δ) 2 ,
where (c, d) and (e, f ) denote the first and the last connected component of

C \ C < i . Next, for C ∈ C η \ C ≥ η define j(C) = max{j ∈ {1, . . . , N } | inf C ϕ j < δ}.
Summarizing the previous estimates we obtain

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I \ (I η ∪ C ≥ )) ≥ C∈Cη\C ≥ η C∩K ε 0 ω ε α 0 , γ ε ε , |ϑ| dx + N i=1 C ≥ i ϕ 2 i + α 2 i ε 2 /β i 2ε |ϑ| 2 dx + C < i ϕ 2 i + α 2 i ε 2 /β i 2ε |ϑ| 2 + β i 2ε (ϕ i -1) 2 dx + C\C < i β i 2 ε|ϕ i | 2 + (ϕ i -1) 2 ε dx ≥ C∈Cη\C ≥ η (η -δ) 2 α 0 m C∩K ε 0 + N i=1 α j(C) m C ≥ i - α 2 j(C) ε 2δ 2 H 1 (C ≥ i ) + α i (1 -δ)m C < i + N i=1 inf C ϕ i ≤δ β i (η -δ) 2 ≥ C∈Cη\C ≥ η (η -δ) 2 (α j(C) m C + β j(C) ) - α 2 j(C) ε 2δ 2 H 1 (C) ≥ (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx - α 2 1 ε 2δ 2 H 1 (I) .
Finally, we obtain the desired estimate,

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) = G ε (ϑ, ϕ 1 , . . . , ϕ N ; I η ∪ C ≥ ) + G ε (ϑ, ϕ 1 , . . . , ϕ N ; I \ (I η ∪ C ≥ )) ≥ (η -δ) 2 Iη∪C ≥ α 0 |ϑ| dx - α 2 0 εH 1 (I) 2δ 2 + (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx - α 2 1 ε 2δ 2 H 1 (I) .
2. (α 0 = ∞) In this case the set K ε 0 is empty, and the cost functional reduces to

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) = I γ ε (x)|ϑ(x)| 2 2ε + N i=1 β i 2 ε|ϕ i (x)| 2 + (ϕ i (x) -1) 2 ε dx.
Chapter 4. Piecewise affine cost functions With Jensen's inequality we thus compute

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I η ∪ C ≥ ) ≥ Iη∪C ≥ γ ε 2ε |ϑ| 2 dx ≥ δ 2 2ε Iη∪C ≥ |ϑ| 2 dx ≥ δ 2 2ε Iη∪C ≥ |ϑ| dx 2 H 1 (I η ∪ C ≥ ) ≥ δ 2 2ε Iη∪C ≥ |ϑ| dx 2 H 1 (I) .
Furthermore, the same calculation as in the case α 0 < ∞ yields

G ε ( θ, ϕ 1 , . . . , ϕ N ; I \ (I η ∪ C ≥ )) ≥ ≥ (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx - α 2 1 ε 2δ 2 H 1 (I)
so that we obtain the desired estimate with respect to weak- * convergence in M(J) and strong convergence in L 1 (J) N .

G ε (ϑ, ϕ 1 , . . . , ϕ N ; I) = G ε (ϑ, ϕ 1 , . . . , ϕ N ; I η ∪ C ≥ ) + G ε (ϑ, ϕ 1 , . . . , ϕ N ; I \ (I η ∪ C ≥ )) ≥ δ 2 2ε Iη∪C ≥ |ϑ| dx 2 H 1 (I) + (η -δ) 2 C∈Cη\C ≥ η max β 1 , h C |ϑ| dx - α 2 1 ε δ 2 H 1 (I) .
Proof. Let (ϑ ε , ϕ ε 1 , . . . , ϕ ε N ) be an arbitrary sequence converging to (ϑ, ϕ 1 , . . . , ϕ N ) in the considered topology as ε → 0, and assume without loss of generality that the limit inferior of G ε [ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; J] is actually a limit and is finite (else there is nothing to show).

It suffices to show the lim inf-inequality for each connected component Ĩ = (a, b)

of J separately. Due to G ε [ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; Ĩ] ≥ β i 2ε ϕ ε i -1 2 L 2 for i = 1, . . . , N we must have ϕ ε i → 1 in L 2 ( Ĩ)
and thus also in L 1 ( Ĩ) so that ϕ i = 1. Even more, after passing to another subsequence, by Egorov's Theorem all ϕ ε i converge uniformly to 1 outside a set of arbitrarily small measure. In particular, for any ξ > 0 we can find an open interval (a + ξ, bξ) ⊂ I ⊂ Ĩ such that ϕ ε i → 1 uniformly on ∂I, and for any η < 1 there is an open set I η ⊂ I with H 1 (I \ I η ) ≤ 1η such that ϕ ε i ≥ η on I η ∪ ∂I for all i = 1, . . . , N and all ε small enough.

We now choose δ = ε 1/3 and η = 1-ε and denote by C η (ε) and C ≥ η (ε) the collections of connected components of I \ I η from Theorem 4.2 (which now depend on ε). Further we abbreviate

C < η (ε) = C η (ε) \ C ≥ η (ε).
The bound of Theorem 4.2 implies that the number of elements in C < η (ε) is bounded uniformly in ε and η. Passing to another 4.3. The Γ-limit of the phase field functional subsequence we may assume C < η (ε) to contain K sets C 1 (ε), . . . , C K (ε) whose midpoints converge to x 1 , . . . , x K ∈ I, respectively. Thus for an arbitrary ζ > 0 we have that for all ε small enough each

C ∈ C < η (ε) lies inside the closed ζ-neighbourhood B ζ ({x 1 , . . . , x K }) of {x 1 , . . . , x K }. Now for α 0 < ∞ we obtain from Theorem 4.2 lim inf ε→0 G ε (ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; I) ≥ lim inf ε→0 (η -δ) 2 Iη∪C ≥ (ε) α 0 |ϑ ε | dx + (η -δ) 2 K i=1 max β 1 , h C i (ε) |ϑ ε | dx ≥ lim inf ε→0 (η -δ) 2 I\B ζ ({x 1 ,...,x K }) α 0 |ϑ ε | dx + (η -δ) 2 K i=1 h B ζ ({x i }) |ϑ ε | dx ≥ α 0 |ϑ|(I \ B ζ ({x 1 , . . . , x K })) + K i=1 h (|ϑ|(B ζ ({x i }))) ,
where in the second inequality we used

A∪B α 0 |ϑ| dx + h C |ϑ| dx ≥ A α 0 |ϑ| dx + h B |ϑ| dx + h C |ϑ| dx ≥ A α 0 |ϑ| dx + h
G ε (ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; I) ≥ α 0 |ϑ|(I \ {x 1 , . . . , x K }) + K i=1 h(|m ϑ (x i )|) ≥ α 0 |ϑ ⊥ |(I) + S ϑ ∩I h(|m ϑ |) dH 0 = G (ϑ; I) .
If on the other hand α 0 = ∞ we obtain from Theorem 4.2

lim inf ε→0 G ε (ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; I) ≥ lim inf ε→0 δ 2 2εH 1 (I) Iη∪C ≥ (ε) |ϑ ε | dx 2 ≥ lim inf ε→0 δ 2 2εH 1 (I) I\B ζ ({x 1 ,...,x K }) |ϑ ε | dx
C i (ε) |ϑ ε | dx > 0 for all ε small enough. Indeed, |m ϑ (x i )| ≤ lim inf ε→0 B ζ ({x i }) |ϑ ε | dx = lim inf ε→0 C i (ε) |ϑ ε | dx + B ζ ({x i })\C i (ε) |ϑ ε | dx , where B ζ ({x i })\C i (ε) |ϑ ε |
G ε (ϑ ε , ϕ ε 1 , . . . , ϕ ε N ; I) ≥ lim inf ε→0 δ 2 2εH 1 (I) Iη∪C ≥ (ε) |ϑ ε | dx 2 + (η -δ) 2 K i=1 |m ϑ (x i )|>0 h C i (ε) |ϑ ε | dx ≥ lim inf ε→0 δ 2 2εH 1 (I) K i=1 |m ϑ (x i )|>0 B ζ ({x i })\C i (ε) |ϑ ε | dx 2 + K i=1 |m ϑ (x i )|>0 h B ζ ({x i }) |ϑ ε | dx -α 1 B ζ ({x i })\C i (ε) |ϑ ε | dx ≥ lim inf ε→0 K i=1 |m ϑ (x i )|>0 h B ζ ({x i }) |ϑ ε | dx - α 2 1 εH 1 (I) 2δ 2 ≥ K i=1 |m ϑ (x i )|>0 h (|ϑ|(B ζ ({x i }))) = S ϑ h(|m ϑ |) dH 0 = G (ϑ; I) ,
where in the second inequality we used h(m 1 + m 2 ) ≤ h(m 1 ) + α 1 m 2 for any m 1 > 0, m 2 ≥ 0 and in the third we optimized in

B ζ ({x i })\C i (ε) |ϑ ε | dx.
The proof is concluded by letting ξ → 0 and noting lim inf ξ→0 G (ϑ; I) ≥ G (ϑ; Ĩ).

The Γlim inf inequality

We now prove the desired lim inf-inequality, which will be obtained by slicing.

Proposition 4.3 (Γlim inf of phase field functional). Let µ + , µ -∈ P(Ω). We have

Γ -lim inf ε→0 F ε = E with respect to weak- * convergence in M(Ω; R 2 ) and strong convergence in L 1 (Ω) N .
Proof. Let (σ ε , ϕ ε 1 , . . . , ϕ ε N ) converge to (σ, ϕ 1 , . . . , ϕ N ) in the considered topology. We first extend σ ε and σ to R 2 \ Ω by zero and ϕ ε i and ϕ i to R 2 \ Ω by 1 for i = 1, . . . , N . The phase field cost functional and the cost functional are extended to R 2 in the obvious way (their values do not change). Without loss of generality (potentially after extracting a subsequence) we may assume lim ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ) to exist and to be finite (else there is nothing to show). As a consequence we have ∇ • σ ε = µ ε +µ ε -as well as ∇ • σ = µ +µ -and ϕ 1 ≡ . . . ≡ ϕ N ≡ 1 (since the phase field cost functional is bounded below by N i=1

β i 2ε ϕ ε i -1 2 L 2 ).

The Γ-limit of the phase field functional

Now let A ⊂ R 2 open and bounded; the restriction of the phase field cost functional to a domain A will be denoted F ε (•; A). Choosing some ξ ∈ S 1 , by Fubini's decomposition theorem we have

F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ; A) = ∞ -∞ A ξ,t ω ε α 0 , γ ε (x) ε , |σ ε (x)| + N i=1 β i 2 ε|∇ϕ i | 2 + (ϕ i -1) 2 ε ξ,t dx dt ≥ ∞ -∞ A ξ,t ω ε α 0 , (γ ε ) ξ,t ε , |σ ε ξ,t | + N i=1 β i 2 ε|(ϕ ε i ) ξ,t | 2 + ((ϕ ε i ) ξ,t -1) 2 ε dx dt = ∞ -∞ G ε (σ ε ξ,t , (ϕ ε 1 ) ξ,t , . . . , (ϕ ε N ) ξ,t ; A ξ,t ) dt ,
where F ε is the dimension-reduced phase field energy and for simplicity we identified the domain A ξ,t of the sliced functions with an open subset of the real line. Fatou's lemma thus implies

lim inf ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ; A) ≥ ∞ -∞ lim inf ε→0 G ε (σ ε ξ,t , (ϕ ε 1 ) ξ,t , . . . , (ϕ ε N ) ξ,t ; A ξ,t ) dt .
By assumption, the left-hand side is finite so that the right-hand side integrand is finite for almost all t ∈ R as well. Pick any such t and pass to a subsequence such that lim inf turns into lim. Indeed σ ε ξ,t * σ ξ,t for every ξ and almost all t, as σ ε * σ and Theorem C.1. Thus, Theorem 4.1 on the reduced dimension problem implies lim inf ε→0 F ε (σ ε ξ,t , (ϕ ε 1 ) ξ,t , . . . , (ϕ ε N ) ξ,t ; A ξ,t ) ≥ F (σ ξ,t , (ϕ 1 ) ξ,t , . . . , (ϕ N ) ξ,t ; A ξ,t )

for almost all t ∈ R so that

lim inf ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ; A) ≥ ∞ -∞
G (σ ξ,t , (ϕ 1 ) ξ,t , . . . , (ϕ N ) ξ,t ; A ξ,t ) dt .

For notational convenience let us now define the auxiliary function κ, defined for open subsets A ⊂ R 2 , as

κ(A) = lim inf ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ; A) .
Furthermore, introduce the nonnegative Borel measure

λ(A) = α 0 |σ ⊥ |(A) + Sσ∩A h(m σ ) dH 1
as well as the |σ|-measureable Borel functions

ψ j : R 2 → R, ψ j = σ |σ| • ξ j
for some sequence ξ j , j ∈ N, dense in S 1 .

Since σ is a divergence measure vector field, we have

κ(A) ≥ ∞ -∞ G (σ ξ j ,t , (ϕ 1 ) ξ j ,t , . . . , (ϕ N ) ξ j ,t ; A ξ j ,t ) dt = ∞ -∞ α 0 |(σ ξ j ,t ) ⊥ |(A ξ j ,t ) + Sσ ξ j ,t ∩A ξ j ,t h(|m σ ξ j ,t |) dH 0 dt = α 0 |σ ⊥ • ξ j |(A) + Sσ∩A h(|m σ |)|θ σ • ξ j | dH 1 ≥ A ψ j dλ
for all j ∈ N where we used Remark 9 in the last equality. By [Bra98, Prop. 1.16] the above inequality implies

κ(A) ≥ A sup j ψ j dλ
for any open A ⊂ R 2 . In particular, choosing A as the 1-neighbourhood of Ω we obtain

lim inf ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ; Ω) = κ(A) ≥ A sup j ψ j dλ = α 0 |σ ⊥ |(A) + Sσ∩A h(m σ ) dH 1 = E (σ) ,
the desired result.

Equicoercivity

Proof of Theorem 4.2. Due to

C > F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ) ≥ β i 2ε ϕ ε i -1 2
L 2 for all i = 1, . . . , N , we have ϕ ε i → 1 in L 2 (Ω) and thus also in L 1 (Ω). Furthermore, we will show further below that σ ε L 1 = σ ε M is uniformly bounded, which by the Banach-Alaoglu theorem implies existence of a weakly-* converging subsequence (still denoted σ ε ) with limit σ ∈ M(Ω; R 2 ). It is now a standard property of Γ-convergence that, due to the above equicoercivity, any sequence of minimizers of F ε contains a subsequence converging to a minimizer of E .

To finish the proof we show uniform boundedness of σ ε in M(Ω; R 2 ). Indeed, using

ω ε (α 0 , γε(x) ε , |σ(x)|) ≥ α 0 2 |σ(x)| for x ∈ K ε 0 (remember that K ε 0 = ∅ for α 0 = ∞) we obtain σ ε M = N i=0 K ε i |σ ε | dx ≤ 2C α 0 + N i=1 K ε i |σ ε | dx
(the first term is interpreted as zero for α 0 = ∞). Furthermore, by Hölder's inequality we have

K ε i |σ ε | dx 2 ≤ K ε i (ϕ ε i ) 2 + α 2 i ε 2 /β i 2ε |σ ε | 2 dx K ε i 2ε (ϕ ε i ) 2 + α 2 i ε 2 /β i dx ≤ C K ε i 2ε (ϕ ε i ) 2 + α 2 i ε 2 /β i dx .
4.3. The Γ-limit of the phase field functional

Choosing now some arbitrary λ ∈ (0, 1) we can estimate

K ε i 2ε (ϕ ε i ) 2 + α 2 i ε 2 /β i dx = K ε i ∩{ϕ ε i <λ} 2ε (ϕ ε i ) 2 + α 2 i ε 2 /β i dx + K ε i ∩{ϕ ε i ≥λ} 2ε (ϕ ε i ) 2 + α 2 i ε 2 /β i dx ≤ 4 α 2 i (1 -λ) 2 K ε i ∩{ϕ ε i <λ} β i (1 -ϕ ε i ) 2 2ε dx + 2ε λ 2 H 2 (Ω) ≤ 4C α 2 i (1 -λ) 2 + 2ε λ 2 H 2 (Ω) . Summarizing, σ ε M ≤ C 2α 0 + N i=1 4C 2 α 2 i (1-λ) 2 + 2εC λ 2 H 2 (Ω).

The Γlim sup inequality

Proposition 4.4 (Γlim sup of phase field functional). Let µ + , µ -∈ P(Ω) be an admissible source and sink. We have

Γ -lim sup ε→0 F ε = E with respect to weak- * convergence in M(Ω; R 2 ) and strong convergence in L 1 (Ω) N .
Proof. Consider a mass flux σ between the measures µ + and µ -. We will construct a recovery sequence (σ ε , ϕ ε 1 , . . . , ϕ ε N ) such that σ ε * σ and ϕ ε 1 → 1, . . . , ϕ ε N → 1 in the desired topology as ε → 0 as well as lim sup ε→0 F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ) ≤ E (σ, 1, . . . , 1). Without loss of generality we may restrict our attention to fluxes for which

E h (σ) = E (σ, 1, . . . , 1) ≤ C < ∞
since otherwise there is nothing to prove. By [BW17, Def. 2.2 & Prop. 2.32] there exists a sequence

σ j = M j k=1 m k,j θ k,j H 1 Σ k,j
of polyhedral divergence measure vector fields in Ω such that

σ j * σ, ∇ • σ j = µ j + -µ j -, µ j ± * µ ± , E µ j + ,µ j - h (σ j ) → E µ + ,µ - h (σ).
In the above formula and in the sequel we will specify the constraint measure for the energy E h to improve readability. If µ + and µ -are finite linear combinations of Dirac masses (which we have assumed in the case α 0 = ∞), we may even choose µ j ± = µ ± . We will construct the recovery sequence based on those polyhedral divergence measure vector fields. In the following we will use the notation

F ε (σ, ϕ 1 , . . . , ϕ N ) = Ω ω ε α 0 , γ ε (x) ε , |σ(x)| dx + N i=1 β i T ε (ϕ i )
for the phase field cost functional even without divergence constraints.

Step 1. Initial construction for a single polyhedral segment In this step we approximate a single line element m k,j θ k,j H 1 Σ k,j of σ j by a phase field version. To this end we fix j and 0 ≤ k ≤ M j and drop these indices from now on in the notation for the sake of legibility. Without loss of generality we may assume Σ = [0, L] × {0}, m > 0, and θ = e 1 the standard Euclidean basis vector. Set

ῑ = argmin{α i m + β i | i = 0, . . . , N }
to identify the phase field that will be active on Σ (ῑ = 0 means that no phase field is active). We first specify (a preliminary version of) the vector field σ ε . To this end let d Σ denote the distance function associated with Σ and define the width

a ε ῑ = α ῑmε 2β ῑ if ῑ > 0 and a ε ῑ = α 0 mε otherwise (4.5)
over which the vector field will be diffused. We now set

σ ε = m 2a ε ῑ χ {d Σ ≤a ε ῑ } e 1 ,
where χ A shall denote the characteristic function of a set A. Note that this vector field encodes a total mass flux of m that is evenly spread over the a ε ῑ -enlargement of Σ. The corresponding active phase field will be zero in that region. Indeed, consider the auxiliary Cauchy problem

φ = 1 ε (1 -φ), φ(0) = 0,
whose solution φ ε (t) = 1exp -t ε represents the well-known optimal Ambrosio-Tortorelli phase field profile. Then we set ϕ ε i = 0 for all i = ῑ and, if ῑ = 0,

ϕ ε ῑ (x) = φ ε (max{0, d Σ (x) -a ε ῑ }) = 0, if d Σ (x) < a ε ῑ , 1 -exp a ε ῑ -d Σ (x) ε , if d Σ (x) ≥ a ε ῑ .
Let us now evaluate the corresponding phase field cost. In the case ῑ = 0 (which can only occur for α 0 < ∞) we obtain

F ε (σ ε (x), ϕ ε 1 , . . . , ϕ ε N ) = Ω ω ε α 0 , γ ε ε , |σ ε | dx ≤ Ω α 0 |σ ε | + ε p |σ ε | 2 dx = {d Σ ≤a ε 0 } 1 + ε p-1 2α 2 0 2ε dx = (α 0 m L + π α 2 0 m 2 ε)(1 + ε p-1 2α 2 0 ) = α 0 m L + C(m, L)ε q ,
where we abbreviated q = min{1, p -1} > 0 and C(m, L) > 0 denotes a constant depending on m and L. In the case ῑ = 0 we have |σ ε | = β ῑ/(α ῑε) as well as

γ ε = α 2 ῑ ε 2 /β ῑ on the support of |σ ε | so that Ω ω ε α 0 , γ ε ε , |σ ε | dx = α 2 ῑ ε 2 β ῑ β ῑ α ῑε 2 1 2ε + ε p-2 β ῑ α ῑ 2 (2 a ε ῑ L + π (a ε ῑ ) 2 ) = α ῑ 2 mL + C(m, L)ε q .
4.3. The Γ-limit of the phase field functional Furthermore we have T ε (ϕ ε i ) = 0 for i = ῑ and, employing the coarea formula,

β ῑT ε (ϕ ε ῑ ) = β ῑ 2 Ω ε|∇ϕ ε ῑ | 2 + (1 -ϕ ε ῑ ) 2 ε δx+ = β ῑ 2ε 2La ε ῑ + π(a ε ῑ ) 2 + β ῑ 2 ∞ a ε ῑ {d Σ =t} ε|φ ε (t -a ε ῑ )| 2 + (φ ε (t -a ε ῑ ) -1) 2 ε dH 1 (x) dt = β ῑ 2ε 2La ε ῑ + π(a ε ῑ ) 2 + β ῑ ∞ a ε ῑ 1 ε exp 2(a ε ῑ -t) ε (2L + 2πt) dt = α ῑ 2 m + β ῑ L + C(m, L)ε . Summarizing, F ε (σ ε , ϕ ε 1 , . . . , ϕ ε N ) ≤ h(m) L + C(m, L)ε q .
Step 2. Adapting sources and sinks of all polyhedral segments The vector field constructions σ ε k,j from the previous step for each polyhedral segment Σ k,j are not compatible with the divergence constraint associated with the measure σ j , that is,

∇ •   M j k=1 σ ε k,j   = ρ ε * µ j + -µ j -.
We remedy this by adapting the source and sink of each σ ε k,j . Set

r(j) = max k=1,...,M j |m k,j | • α 1 β 1 if α 0 = ∞, max α 1 β 1 , α 0 else,
then all vector fields σ ε k,j have support in a band around Σ k,j of width no larger than r(j)ε. Without loss of generality we assume r(j) ≥ 1 (else we just increase it). Again we concentrate on a single segment with fixed j and k and drop these indices in the following (we will also write r instead of r(j)). Denote by s + and s -the starting and ending point of the segment Σ with respect to the orientation induced by θ. Consider the elliptic boundary value problems

∆u ± (x) = ±m (ρ * δ 0 )(x) on B r (0), ∇u ± (x) • ν(x) = εσ ε (εx + s ± ) • ν(x) on ∂B r (0),
where δ y denotes a Dirac mass centered at y, B r (y) denotes the open ball of radius r around y, and ν denotes the outer unit normal to ∂B r (0). Note that the boundary value problems and their solutions u + and u -are independent of ε due to the definition of σ ε . Setting

σε (x) = ∇u ± ((x -s ± )/ε)/ε if x ∈ B εr (s ± ), σ ε (x) else,
(where we assume ε small enough such that B εr (s + ) and B εr (s -) do not intersect) it is straightforward to check

∇ • (σ ε ) = m ρ ε * (δ s --δ s -) .
Furthermore, to have at least one phase field zero on the new additional support B εr (s + ) ∪ B εr (s -) of the vector field we set and (ϕ ε i ) j = min{(ϕ ε i ) j,1 , . . . , (ϕ ε i ) j,M j } for i = 1, . . . , N . Obviously, we have, as desired,

ϕ ε 1 (x) = min ϕ ε 1 (x), P (|x -s -|), P (|x -s + |) with P (t) = 0 if t ≤ r ε,
∇ • σj = ∇ • (ρ ε * σ j ) = ρ ε * µ j + -µ j -.
Let us now estimate the costs. Let us assume that ε is small enough so that all balls B εr(j) (s ± k,j ) are disjoint as are the supports supp σε k,j \ (B εr(j) (s + k,j ) ∪ B εr(j) (s - k,j )) for all k. An upper bound can then be achieved via

F ε (σ ε j , (ϕ ε 1 ) j , . . . , (ϕ ε N ) j ) ≤ M j k=1 F ε [σ ε j,k , (ϕ ε 1 ) k,j , . . . , (ϕ ε N ) k,j ] + + B εr(j) (s + j,k )∪B εr(j) (s - j,k ) ω ε α 0 , (γ ε ) j ε , |σ ε j | dx + + β 1 2 (B 2εr(j) (s + j,k )∪B 2εr(j) (s - j,k ))∩{(ϕ ε 1 ) j <(ϕ ε 1 ) j } ε|∇(ϕ ε 1 ) j | 2 + 1 ε ((ϕ ε 1 ) j -1) 2 dx .
4.3. The Γ-limit of the phase field functional

The last summand can be bounded above by

β 1 B 2εr(j) (0) ε|P (|x|)| 2 + 1 ε (P (|x|) -1) 2 dx ≤ Cε
for some constant C > 0. For the second summand, note that (γ ε ) j ≤ α 2 1 ε 2 /β 1 on B εr(j) (s ± k,j ) due to (ϕ ε 1 ) j = 0 there; furthermore,

ω ε (α 0 , (γ ε ) j /ε, |σ ε j |) ≤ (γ ε ) j 2ε + ε p |σ ε j | 2 ≤ α 2 1 ε 2β 1 + ε p |σ ε j | 2 .
Thus, if we set S ± = {l ∈ {1, . . . , M j } | s ± j,l = s} for fixed s = s + k,j or s = s - k,j we have

B εr(j) (s) ω ε α 0 , (γ ε ) j ε , |σ ε j | dx ≤ Cε B εr(j) (s) |σ ε j | 2 dx = Cε B εr(j) (s) l∈S + ∇u + j,l ( x-s ε ) ε + l∈S - ∇u - j,l ( x-s ε ) ε 2 dx ≤ CεM j   l∈S + B εr(j) (s) ∇u + j,l ( x-s ε ) ε 2 dx + l∈S -B εr(j) (s) ∇u - j,l ( x-s ε ) ε 2 dx   = CεM j l∈S + B r(j) (0) ∇u + j,l (x) 2 dx + l∈S -B r(j) (s) ∇u - j,l (x) 2 dx = C(s, σ j )ε
for some constant C(σ j ) > 0 depending on the polyhedral divergence measure vector field σ j and the considered point s. In summary, we thus have

F ε (σ ε j , (ϕ ε 1 ) j , . . . , (ϕ ε N ) j ) ≤ ≤ M j k=1 h(m j,k )H 1 (Σ j,k ) + C(m j,k , L j,k )ε q + C(s + j,k , σ j )ε + C(s - j,k , σ j )ε + Cε ≤ E µ j + ,µ j - h (σ j ) + C(σ j )ε q
for some constant C(σ j ) depending on σ j .

Step 3. Correction of the global divergence Recall that the vector field σ ε to be constructed has to satisfy

∇ • σ ε = ρ ε * (µ + -µ -).
In the case α 0 = ∞ the vector field σε j already has that property due to µ j ± = µ ± (thus we set σ ε j = σε j ). However, if α 0 < ∞ (so that admissible sources and sinks µ + and µ -do not have to be finite linear combinations of Dirac masses) we still need to adapt the vector field to achieve the correct divergence. To this end, let λ j ± ∈ M({x ∈ Ω | dist(x, ∂Ω) ≥ ε; R 2 ) be the optimal Wasserstein-1 flux between µ j ± and µ ± , that is, λ ± j minimizes λ M among all vector-valued measures λ with

∇ • λ = µ j ± -µ ± . Setting σ ε j = σε j + ρ ε * (λ j + -λ j -)
we thus obtain ∇ • σ ε j = ρ ε * (µ +µ -), as desired. The additional cost can be estimated using the fact

ω ε α 0 , (γ ε ) j ε , |a + b| ≤ ω ε α 0 , (γ ε ) j ε , |a| + α 0 |b| + ε p (a 2 + 2b 2 )
as well as

ρ ε * λ j ± L ∞ ≤ C µ j + M + µ j -M ε
, where µ j ± M is an upper bound for the total mass moved by λ j ± and the constant C > 0 depends on ρ. With those ingredients we obtain

F ε (σ ε j , (ϕ ε 1 ) j , . . . , (ϕ ε N ) j ) ≤ F ε [σ ε j , (ϕ ε 1 ) j , . . . , (ϕ ε N ) j ] + α 0 ρ ε * (λ j + -λ j -) L 1 + ε p ( σε j 2 L 2 + 2 ρ ε * (λ j + -λ j -) 2 L 2 ) . Now ρ ε * (λ j + -λ j -) L 1 ≤ λ j + -λ j -M ≤ λ j + M + λ j -M = W 1 (µ j + , µ + )+W 1 (µ j -, µ -) = κ j for a constant κ j > 0 satisfying κ(σ j ) → 0 as j → ∞
since the Wasserstein-1 distance W 1 (•, •) metrizes weak- * convergence. Furthermore, in the previous steps we have already estimated

ε p σε j 2 L 2 ≤ C(σ j )ε q . Finally, ε p ρ ε * (λ j + -λ j -) 2 L 2 ≤ ε p ρ ε * (λ j + -λ j -) L ∞ ρ ε * (λ j + -λ j -) L 1 ≤ 2Cε p-1 ( µ j + M + µ j -M )κ j .
Summarizing,

F ε (σ ε j , (ϕ ε 1 ) j , . . . , (ϕ ε N ) j ) ≤ E µ j + ,µ j - h (σ j )+C(σ j )ε q +α 0 κ j +2Cε p-1 ( µ j + M + µ j -M )κ j ≤ E µ j + ,µ j - h (σ j ) + Cκ j + C(σ j )ε q
for some constant C > 0 and C(σ j ) > 0 depending only on σ j .

Step 4. Extraction of a diagonal sequence We will set

σ ε = σ j (ε) ε , ϕ ε 1 = (ϕ ε 1 ) j(ε) , . . . , ϕ ε N = (ϕ ε N ) j(ε)
for a suitable choice j(ε). Indeed, for a monotonic sequence ε 1 , ε 2 , . . . approaching zero we set j(ε 1 ) = 1 and

j(ε i+1 ) = j(ε i ) if C(σ j(ε i )+1 )ε q i > 1 j(ε i )+1 , j(ε i ) + 1 else. Then j(ε i ) → ∞ and C(σ j(ε i ) )ε q i → 0 as i → ∞ so that F ε i (σ ε i , ϕ ε i 1 , . . . , ϕ ε i N ) = F ε i (σ ε i j(ε i ) , (ϕ ε i 1 ) j(ε i ) , . . . , (ϕ ε i N ) j(ε i ) ) ≤ E µ j(ε i ) + ,µ j(ε i ) - h (σ j(ε i ) ) + Cκ j(ε i ) + C(σ j(ε i ) )ε i q ≤ E µ j(ε i ) + ,µ j(ε i ) - h (σ j(ε i ) ) + Cκ j(ε i ) + 1 j(ε i ) → E µ + ,µ - h (σ) = E h (σ).

Numerical experiments

Numerical experiments

Here we describe the numerical discretization with finite elements and the subsequent optimization procedure used in our experiments.

Discretization

The proposed phase field approximation allows a simple numerical discretization with piecewise constant and piecewise linear finite elements. Let T h be a triangulation of the space Ω of grid size h such that Ω = T ∈T h T . Denoting by P m the space of polynomials of degree m, we define the finite element spaces

X 0 h = {v h ∈ L ∞ (Ω) | v h T ∈ P 0 ∀ T ∈ T h } , X 1 h = {v h ∈ C 0 (Ω) | v h T ∈ P m ∀ T ∈ T h } .
Using as before the notation

F ε (σ, ϕ 1 , . . . , ϕ N ) = Ω ω ε α 0 , γ ε (x) ε , |σ(x)| dx + N i=1 β i T ε (ϕ i ) ,
the discretized phase field problem now reads min

(σ,ϕ 1 ,...,ϕ N )∈X 0 h ×(X 1 h ) N ϕ 1 ∂Ω =...=ϕ N ∂Ω =1 F ε (σ, ϕ 1 , . . . , ϕ N ) such that Ω -σ • ∇λ dx = Ω ρ ε * (µ + -µ -)v h dx ∀λ ∈ X 1 h ,
where all integrals are evaluated using midpoint quadrature and the divergence constraint is enforced in weak form. In our numerical experiments we use Ω = (0, 1) 2 with a regular quadrilateral grid whose squares are all divided into two triangles.

Optimization

Here we describe the numerical optimization method used to find a minimizer of the objective functional. We first elaborate the simplest case in which the transport cost h only features a single affine segment, that is, α 0 = ∞ and N = 1. Afterwards we consider the setting with N > 1 and subsequently also with α 0 < ∞, which requires more care due to the higher complexity of the energy landscape.

Single phase field and no diffuse mass flux (N = 1, α 0 = ∞) In this case the energy reads

F ε (σ, ϕ) = Ω γ ε (x) ε |σ(x)| 2 2 + β 1 2 ε|∇ϕ(x)| 2 + (ϕ(x) -1) 2 ε dx with γ ε (x) = ϕ(x) 2 + α 2 1 ε 2 /β 1 .
The employed optimization method is similar to the one presented in [START_REF] Chambolle | Variational approximation of size-mass energies for k-dimensional currents[END_REF] and updates the variables σ and ϕ alternatingly.

Let us abbreviate f ε = ρ ε * (µ +µ -). For minimization with respect to σ, we use the dual variable λ ∈ X 1 h to enforce the divergence constraint and write min

σ∈X 0 h Ω σ•∇λ+λfε dx=0 ∀λ∈X 1 h Ω γ ε ε |σ| 2 2 dx = min σ∈X 0 h max λ∈X 1 h Ω γ ε ε |σ| 2 2 -σ • ∇λ -λf ε dx = max λ∈X 1 h min σ∈X 0 h Ω γ ε ε |σ| 2 2 -σ • ∇λ -λf ε dx ,
where the last step follows by standard convex duality. The minimization in σ can be performed explicitly, yielding σ = ε∇λ γε . Inserting this solution leads to a maximization problem in λ,

max λ Ω - ε|∇λ| 2 2γ ε -λf ε dx .
The corresponding optimality conditions,

Ω ε∇λ • ∇µ γ ε dx = - Ω µf ε dx ∀µ ∈ X 1 h , (4.6)
represent a linear system of equations that can readily be solved numerically for λ so that subsequently σ can be computed (note that Ω f ε dx = 0 so that the above equation has a solution). Fixing σ, the optimality condition for ϕ reads

Ω |σ| 2 ϕψ ε + β 1 ε∇ϕ • ∇ψ + β 1 ε (ϕ -1)ψ dx = 0 ∀ψ ∈ X 1 h with ψ ∂Ω = 0 , (4.7)
which can again be solved under the constraint ϕ 1 ∂Ω = 1 by a linear system solver.

In addition to the alternating minimization a stepwise decrease of the phase field parameter ε is performed, starting from a large value ε start for which the energy landscape shows fewer local minima. Algorithm 4 summarizes the procedure.

Algorithm 4 Minimization for

N = 1, α 0 = ∞ function SPFS(ε start , ε end , N iter , α 1 , β 1 , µ + , µ -, ρ ε end ) set f ε = (µ + -µ -) * ρ ε end , σ 0 = 0 for j = 1, . . . , N iter do set ε j = ε start -(j -1) εstart-ε end N iter -1
set ϕ j to the solution of (4.7) for given fixed σ = σ j-1 set γ j ε = (ϕ j ) 2 + α 2 1 ε 2 j /β 1 set λ j to the solution of (4.6) for given fixed

γ ε = γ j ε set σ j = ε j ∇λ j 2γ j ε end for end function return σ N iter , ϕ N iter , λ N iter

Numerical experiments

Multiple phase fields and no diffuse mass flux (N > 1, α 0 = ∞) Again we aim for an alternating minimization scheme. Fixing ϕ 1 , . . . , ϕ N , the optimization for σ is the same as before, since only γ ε changes. However, the optimization in the phase fields ϕ 1 , . . . , ϕ N is strongly nonconvex (due to the minimum in γ ε ) and thus requires a rather good initialization and care in the alternating scheme.

To avoid minimization for phase field ϕ i with the min-function inside γ ε we perform a heuristic operator splitting: in each iteration of the optimization we first identify at each location which term inside

γ ε = min(ϕ 2 1 + α 2 1 ε 2 /β 1 , . . . , ϕ 2 N + α 2 N ε 2 /β N )
is the minimizer, which is equivalent to specifying the regions

R ε i = {x ∈ Ω | γ ε (x) = ϕ i (x) 2 + α 2 i ε 2 /β i } , i = 1, . . . , N. (4.8)
Afterwards we then optimize the energy F ε separately for each phase field ϕ i assuming the regions R ε i fixed, that is, we minimize

N i=1 Ω ϕ i (x) 2 + α 2 i ε 2 /β i ε |σ(x)| 2 2 χ R ε i (x) + β i 2 ε|∇ϕ i (x)| 2 + β i 2 (ϕ i (x) -1) 2 ε dx ,
where χ R ε i is the characteristic function of region R ε i . Similarly to the case N = 1 of a single phase field, this amounts to solving the linear system

Ω ϕ i ψ ε |σ| 2 χ R ε i + β i ε∇ϕ i • ∇ψ + β i ϕ i ψ ε dx ∀ψ ∈ X 1 h with ψ ∂Ω = 0 (4.9)
for ϕ i ∈ X 1 h with ϕ i ∂Ω = 1. Since in the above simple approach the regions R ε i and phase fields ϕ i can move, but cannot nucleate within a different region (indeed, imagine for instance R ε 1 = Ω, then ϕ 2 , . . . , ϕ N will be optimized to equal 1 everywhere so that in the next iteration again R ε 1 = Ω), a suitable initial guess is crucial. To provide such a guess for the initial regions R ε i , we proceed as follows. We first generate some flux network σ 0 consistent with the given source and sink. This can for instance be done using the previously described algorithm for just a single phase field: in our simulations we simply ignore ϕ 2 , . . . , ϕ N and pretend only the phase field ϕ 1 would exist (essentially this means we replace the cost function h by m → α 1 m + β 1 for m > 0; of course, an alternative choice would be to take m → αm + β for some α, β > 0 that better approximate h for a larger range of values m). We then identify the total mass flowing through each branch of σ 0 . To this end we convolve |σ 0 | with the characteristic function χ Br(0) of a disc of radius r. If r is sufficiently large compared to the width of the support of σ 0 , we obtain

χ Br(0) * |σ 0 | (x) = Br(x) |σ 0 |(y) dy ≈ 2rm(x) ,
where m(x) denotes the mass flux through the nearby branch of σ 0 . Now we can compute the regions

R ε i = {x ∈ Ω | i = argmin j=1,...,N {α j m(x) + β j }} (4.10)
and furthermore use σ 0 as initial guess of the vector field. Algorithm 5 summarizes the full procedure.

Chapter 4. Piecewise affine cost functions

Algorithm 5 Minimization for N > 1, α 0 = ∞ function MPFS(ε start , ε end , N iter , α 1 , . . . , α N , β 1 , . . . , β N , µ + , µ -, ρ ε end ) set f ε = (µ + -µ -) * ρ ε end set (σ 0 , •, •) = SP F S(ε start , ε end , N iter , α 1 , β 1 , µ + , µ -, ρ ε end ) compute regions R ε 1 , . . . , R ε N via (4.10) for j = 1, . . . , N iter do set ε j = ε start -(j -1) εstart-ε end N iter -1
set ϕ j i to the solution of (4.9) for given fixed σ = σ j-1 , i = 1, . . . , N update regions R ε 1 , . . . , R ε N via (4.8) set γ j ε = min i=1,...,N (ϕ j i ) 2 + α 2 i ε 2 /β i set λ j to the solution of (4.6) for given fixed

γ ε = γ j ε set σ j = ε j ∇λ j 2γ j ε end for end function return σ N iter , ϕ N iter 1 , . . . , ϕ N iter N
Note that the estimate m(x) of the flowing mass is only valid in close proximity of the support of σ 0 so that the regions R ε i are only reliable near σ 0 . However, away from σ 0 all phase fields will be close to 1 anyway so that the regions R ε i do not play a role there. The effectiveness of the initialization can be further improved by an energy rescaling which we typically perform in our simulations: Recall that the optimal width (4.5) of the support of the vector field σ not only depends on the transported mass m(x), but also on which phase field is active at x. Thus, initializing with some vector field σ that was computed based on preliminary active phase fields may erroneously give slight preference to incorrect phase fields. This can be avoided by a small parameter change which assigns a different ε to each phase field. Indeed, setting ε i = β i ε/α i to be the phase field parameter associated with phase field ϕ i , equation (4.5) shows that the support width of σ becomes mε and thus no longer depends on the phase field. Thus, in practice we usually work with the phase field cost

Fε (σ, ϕ 1 , . . . , ϕ N ) = Ω ω ε α 0 , γε (x) ε , |σ(x)| dx + N i=1 β i T ε i (ϕ i ) (4.11) with γε (x) = min i=1,...,N {ϕ i (x) 2 + α 2 i εε i /β i } = min i=1,...,N {ϕ i (x) 2 + α i ε 2 }.
The Γconvergence result can readily be adapted to this case.

Multiple phase fields and diffuse mass flux (N > 1, α 0 < ∞) The difference to the previous case is that now there may be nonnegligible mass flux σ in regions where no phase field ϕ 1 , . . . , ϕ N is active. Correspondingly, we adapt the previous alternating minimization scheme by introducing the set

R ε 0 = x ∈ Ω | |σ(x)| > α 0 γ ε (x)/ε ,
which according to the form of ω ε in equation (4.1) describes the region in which mass flux σ is penalized by α 0 |σ|. The regions in which the i th phase field is active are thus

4.4. Numerical experiments modified to Rε i = R ε i \ R ε 0 . (4.12)
As before, we now separately minimize

N i=1 Ω ϕ i (x) 2 + α 2 i ε 2 /β i ε |σ(x)| 2 2 χ Rε i (x) + β i 2 ε|∇ϕ i (x)| 2 + β i 2 (ϕ i (x) -1) 2 ε dx , (4.13)
for each phase field ϕ i . The optimization for σ changes a little compared to the previous case since the problem is no longer quadratic and thus no longer reduces to solving a linear system. Instead we will perform a single step of Newton's method in each iteration. The optimization problem in σ reads min

σ∈X 0 h Ω σ•∇λ+λfε dx=0 ∀λ∈X 1 h Ω ω ε α 0 , γ ε ε , |σ| dx ,
and its optimality conditions are

0 = Ω ξ(|σ|)σ • ψ -∇λ • ψ δx for all ψ ∈ X 0 h , 0 = Ω σ • ∇µ + µf ε dx for all µ ∈ X 1 h , (4.14)
where

ξ(|σ|) = 1 |σ| ∂ 3 ω ε α 0 , γ ε ε , |σ| = min γ ε ε , α 0 |σ| + 2ε p .
Letting σ and λ be the coefficients of σ and λ in some basis {b 0 i } i of X 0 h and {b 1 i } i of X 1 h , respectively, the optimality conditions can be expressed as

0 = R(σ, λ) = M [ξ(|σ|)] B B T 0 σ λ + 0 F ,
where the finite element matrices and vectors are defined as

M [ξ] ij = Ω ξb 0 i b 0 j dx , B ij = Ω b 0 i ∇b 1 j dx , F i = Ω b 1 i f ε dx .
In each iteration of the alternating minimization scheme we now take one Newton step for 0 = R(σ, λ). As before, the algorithm requires a suitable initial guess, which is determined in the same way as for the case without diffuse component. Algorithm 6 summarizes the alternating scheme.

Experimental results

The algorithms were implemented in MATLAB c ○ ; parameters reported in this section refer to the rescaled cost (4.11). We first present simulation results for a single phase field and no diffuse mass flux, N = 1 and α 0 = ∞. Figure 4.5 shows simulation results for the same source and sink configuration as in Figure 4.4, but this time with N = 3 different linear segments in h and corresponding phase fields. It is clear that different phase fields become active on the different network branches according to the mass flux through each branch. This can be interpreted as having streets of three different qualities: the street ϕ 3 allows faster (cheaper) transport, but requires more maintenance than the others, while street ϕ 1 requires the least maintenance and only allows expensive transport.

The case α 0 < ∞ finally can be interpreted as the situation in which mass can also be transported off-road, that is, part of the transport may happen without a street network, thus having maintenance cost β 0 = 0, but at the price of large transport expenses α 0 per unit mass. Corresponding results for again the same source and sink configuration are shown in Figure 4.6. In contrast to the case α 0 = ∞ it is now also possible to have sources and sinks that are not concentrated in a finite number of points. A corresponding example is shown in

Chapter 5

Generalized cost functions

Introduction

In the previous chapters we have considered the phase field function (1ϕ) 2 to model the characteristic function of the support of the vector valued measure σ. The main idea behind this approach was, for a weakly differentiable function ϕ, to consider the measure

µ ϕ (A) = Ω∩A |∇ϕ| |1 -ϕ| dx = Ω∩A |∇W (ϕ)| dx.
where A is any Borel set and W (t) = t 2 /2. As a matter of fact, given a sequence of functions ϕ ε for which the quantity

Ω ε|∇ϕ ε | 2 + (1 -ϕ ε ) 2 ε dx
is bounded independently of ε then the sequence of Radon measures µ ϕε weakly- * converges up to subsequence to the measure H n-1 {ϕ = 1}. As already stated in the introduction of this thesis, this fact is essential when approximating the Mumford -Shah functional as in the limit energy one aims at recovering the length of the jump set of some BV function which is contained in the set {ϕ = 1}. Later on some more general functionals have been introduced with different penalization of the jump set [START_REF] Bouchitté | Relaxation results for some free discontinuity problems[END_REF] to model fractures. Recently [ABS99, DMOT16] other phase-field methods have dealt with the problem of efficiently approaching these energies. The main idea is that the limit ϕ rather then acquiring only the values {0, 1} should range over [0, +∞). In this chapter we follow this method to approximate any functional E h where h is a concave cost function.

A way to approximate an energy E h is to substitute for h a sequence h k of piecewise affine functions pointwise converging to h and apply the method described in Chapter 4. The quality of such approximation depends on the number of phase fields used and for an accurate approximation, the numerical complexity of the method may become prohibitive. Here we propose a different model with a single phase function ϕ its main drawback is that the term in the energy that penalizes σ is linear in σ, it was quadratic or at least strictly convex in the preceding models of the thesis. Let us be more formal. Let us consider a convex open set Ω ⊂ R 2 . We let µ -, µ + ∈ P(Ω) and denote by X ε the subset of M(Ω, R 2 ) × L 2 (Ω) of those pairs (σ, ϕ) such that

∇ • σ = (µ + -µ -) * ρ ε for a standard symmetric convolution kernel ρ ε . For a couple (σ, ϕ) ∈ M(Ω) × L 2 (Ω) we set F ε (σ, ϕ) :=    Ω f (ϕ)|σ| + 1 2 ε|∇ϕ| 2 + ϕ 2 ε dx if (σ, ϕ) ∈ X ε +∞ otherwise.
where the function f : R → R is defined as

f (t) := (-h * ) -1 (t 2 ).
(5.1)

In the above formula h * is the concave Legendre transform of h (see Section 5.2 for its precise definition). The limit energy E is defined in equation (4.2) of Chapter 4. We prove:

Theorem 5.1. Let h : R → [0, +∞) be an even, continuous function such that h(0) = 0 and h is concave on [0, +∞).

(5.2)

Let f be defined as above then

F ε Γ - → E (5.3)
as ε → 0.

In Section 5.2 we recall the definition of Legendre transform for a concave function and obtain some properties of the function f which are essential to the Γ-convergence result. The convergence result is obtained again by slicing and we will take advantage of the results in Appendix C following a strategy similar to the one in Chapter 4. We introduce the reduced dimension problem and study the upper and lower bound for the Γ-convergence result in Section 5.3.

Origin of the model and preliminaries

Let us give a brief idea of the model. Let σ = (m, τ, Σ) be a rectifiable vector measure so that the energy may be written as

E h (σ) = Σ h(m(x)) dH 1 .
Now recall that for a concave function its Legendre transform is defined as

h * (z) := inf m {z m -h(m)}.
Furthermore by [ABM14, Theorem 9.3.2] it holds h * * := (h * ) * = h thus we may write

E h (σ) = Σ inf{z m(x) -h * (z)} dH 1 .

Origin of the model and preliminaries

Now letting z be a function we can interchange the integral and the inf signs obtaining

E h (σ) = inf z Σ z(x) m(x) -h * (z(x)) dH 1 .
In the above formula we may notice the presence of two measures supported on the rectifiable set Σ with H 1 Σ-density respectively z(x) m(x) and h * (z(x)). We now model our approximating functional. The main idea is to retrieve the measure h * (z(x)) H 1 Σ by means of a phase field approach. Contrary to the previous approaches we now suppose that the phase field ϕ takes value 0, not 1, outside an ε-neighborhood of Σ and some value ϕ(x)

∈ [0, 1] if x ∈ Σ. Given a potential W : R → R + we let c W (t) := 2 |t| 0 W (s) d s (5.4)
be the cost of the transition between 0 and a value t. We set -h * (z(x)) = c W (ϕ(x)) and suppose that h * is invertible so that, setting f (ϕ(x)

) := (-h * ) -1 • c W (ϕ(x))
, by a change of variables we have

E h (σ) = inf ϕ Σ f (ϕ(x)) m(x) + c W (ϕ(x)) dH 1 .
Let us observe that the first addend in the latter corresponds to Ω f (ϕ(x)) d|σ|. Furthermore by reversing the Modica-Mortola arguments used in the previous chapters when dealing with the ϕ components we know that, up to a small error,

Σ c W (ϕ(x)) dH 1 is equal to Ω 1 2 ε|∇ϕ| 2 + W (ϕ)
ε dx. Considering as potential the function W (x) = x 2 and replacing σ with a mollified version of itself we are led to the proposed approximating functional, namely

F ε (σ, ϕ) = Ω f (ϕ(x))|σ| + 1 2 ε|∇ϕ| 2 + ϕ 2 ε dx.
(5.5)

Let us specify what we will consider when talking about the Legendre transform of h. Since h is an even function first of all consider its restriction to [0, +∞). Define the quantities lim

m 0 + h(m) m = α 0 > 0, lim m +∞ h(m) m = α ∞ ≥ 0 and lim m +∞ (α ∞ m -h(m)) = β ∞ .
Being h concave and non decreasing we have inf

m∈[0,+∞) {m z -h(m)} = -∞, for z < α ∞ , 0, for z ≥ α 0 .
The first fact follows easily from the inequality h(m) ≤ α 0 m. For the second observe that for any m and t ≥ m we have

h(m) = h m t t + 1 - m t 0 ≥ m t h(t) + 1 - m t h(0)
thus passing to the limit as t → +∞ we obtain h(m) ≥ α ∞ m. We call Legendre transform the function

h * (z) := inf m∈[0,∞) {m z -h(m)} .
In the following we will always consider the restriction of h * on the interval in which is well defined and finite, namely properties for h.

h * : [α ∞ , α 0 ] → [-β ∞ , 0]. In the case α ∞ = ∞ or β ∞ = ∞
Lemma 5.1 (Properties for h * ). We have:

1. h * is continuous, 2. h * is concave, 3. h * is non decreasing, 4. h * (α 0 ) = 0.
Proof. The function h * is continuous and concave since is the infimum of a family of affine functions. Let us prove that h * is monotone non decreasing. By contradiction suppose the existence of two values z 1 < z 2 such that h * (z 1 ) > h * (z 2 ). Therefore there exists an ε > 0 such that for any x ∈ [0, ∞) it holds

z 1 x -h * (z 1 ) + ε < z 2 x -h * (z 2 ). Now let x ε be such that z 2 x ε -h(x ε ) < h * (z 2 ) + ε/2 so we obtain h(x ε ) + ε ≤ z 1 x ε -h * + ε < z 2 x ε -h * (z 2 ) < h(x ε ) + ε 2 .

Origin of the model and preliminaries

The latter is a contradiction thus h * is monotone non decreasing. Finally, by the inequality h(m) ≤ α 0 we obtain

h * (α 0 ) = inf{α 0 m -h(m)} ≥ 0
and the latter is actually a minimum as it is evident by choosing m = 0.

The properties stated above ensure that -h * defines a bijection between the intervals [α ∞ , α 0 ] and [0, β ∞ ] and may be inverted. Consider the inverse function (-h * ) -1 which, in the case β ∞ < ∞, we extend constant on [0, +∞). Recalling the equation (5.4) we set

f := (-h * ) -1 • c W . (5.6)
From the properties of h * we easily derive:

Lemma 5.2 (Properties for f ). Let W : R → R + be a non negative, increasing for x ≥ 0 and even function such that W (0) = 0 then the function

f := (-h * ) -1 • c W is: a. continuous on [0, ∞), b. non decreasing, c. f ≥ 0 and f (0) = α 0 ,
Furthermore the following identity holds true

inf z∈[0,1] { f (z)m + c W (z) } = h(m).
Before moving to the proof of the Γ-convergence result let us produce some examples of function f . In all these cases we will consider the potential function W (x) = x 2 .

1. The first examples we consider is given by a function with linear growth both at the origin and at in infinity. In facts, for some values α 0 > α 1 > . . . > α N ≥ 0 and 0 ≤ β 0 < β 1 < . . . < β N we consider the piecewise affine functions

h(m) = min{α i m + β i : i ∈ {1, . . . , N }}.
Indeed, we have

lim m↓0 h(m) m = α 0 , α ∞ = α N and β ∞ = β N .
A direct evaluation gives

h * (x) = inf m∈[0∞) {x m -h(m)} =          -∞, x < α N , - β i -β i-1 α i -α i-1 (x -α i ) -β i , α i ≤ x < α i-1 , 0, x ≥ α 0 .
Chapter 5. Generalized cost functions

x h(x) h * (x) α 2 =α ∞ α 1 α 0 β ∞
Thus by our notion of Legendre transform h * is the restriction of the above to the interval

[α N , α 0 ]. Indeed, h * defines a bijection of [α N , α 0 ] onto [-β ∞ , 0]. Since for our choice of W we have c W (x) = x 2 the function f is given by f (x) :=    α i -α i-1 β i -β i-1 (x 2 -β i ) + α i , x ∈ [ β i-1 , β i ), α N , x ≥ β N .
Remark that we have extended (-h * ) -1 on [β N , ∞) with the value α N . 2. The second example is a function with linear growth near at the origin namely let h(m) := 1 + |m| -1. We have

α 0 = 1 2 , α ∞ = 0 and β ∞ = +∞.
For this choice we obtain

inf m∈[0,∞) {x m -h(m)} = 1 -x - 1 4x
which is well defined and invertible on the interval [0, 1/2], namely we have

(-h * ) -1 (x) = x + 1 - √ x 2 + 2x 2 and we set f (x) = x 2 + 1 - √ x 4 + 2x 2 2 .

The third example has linear growth at infinity and is given by

h(m) := m + √ m. α 0 = +∞, α ∞ = 1 and β ∞ = +∞. 5.3. Proof of Theorem 5.1 x h(x) h * (x) x f (x) (0, α 0 ) x h(x) h * (x) x = α ∞ x f (x) y = α 0
In this case the Legendre transform is given by h * (x) = 1/(4 -4x) and the function f may be defined as

f (x) = 1 + 1 4x 2 .

4.

The last example we deal with is the branched transport case. For p > 1 consider the function h(m) = p m 1/p for which we have

α 0 = +∞, α ∞ = 0 and β ∞ = +∞.
A direct evaluation gives h * (x) = (1p) x 1/1-p as show in Figure 5.1. Therefore we have

f (x) = x 2 p -1 1-p .

Proof of Theorem 5.1

This section is devoted to the proof of the Γ-convergence result. Throughout the whole section we will consider the potential W (t) := t 2 for simplicity but the result is valid for a wider class of problems. The Γlim inf inequality is obtained again by slicing.

Chapter 5. Generalized cost functions

x f (x)
For this reason following the strategy of Chapter 4, Section 4.3 we define the reduced dimension problem. Given an interval I and a measure θ ∈ M(I) we recall that it may be decomposed into its atomic component and diffuse one so that

θ = θ ⊥ + m θ H 0 S θ
where S θ is a countable set of points. Analogously to what has been already done in the previous chapter let us introduce the functional G :

M(I)×L 2 (I) → [0, ∞) defined as G (θ, ϕ; I) =    h (0)|θ ⊥ | + S θ h(m θ ) dH 0 , if ϕ = 0, +∞, otherwise.
We introduce as well the the reduced phase field functional

G ε (θ, ϕ; I) :=    I f (ϕ)|θ| + 1 2 ε|ϕ | 2 + ϕ 2 ε dx, for (θ, ϕ) ∈ L 1 (I) × L 2 (I), +∞, otherwise on M(I) × L 2 (I).
We prove that

Lemma 5.3 (Γ -lim inf reduced inequality). Let I ⊂ R be an open set, (θ, ϕ ε ) ∈ M(I) × L 2 (I). For any (θ ε , ϕ ε ) such that θ ε * θ and ϕ ε → ϕ it holds lim inf ε↓0 G ε (θ ε , ϕ ε ; I) ≥ G (θ, ϕ; I).
Proof. With no loss of generality we may assume that I is an interval, that for every

ε > 0, θ ε ∈ L 1 (I), ϕ ε ∈ W 1,2 (I) and lim inf ε↓0 G ε (θ ε , ϕ ε ; I) ≤ M < +∞
otherwise the inequality is trivial. We further assume that the decomposition for the limit measure θ reads θ = θ ⊥ + j∈N m j δ p j .

Since we have assumed the family (θ ε , ϕ ε ) to be equibounded in energy it holds

Ω ϕ 2 ε dx ≤ εM.
Therefore, up to a subsequence, ϕ ε → 0 pointwise almost everywhere. Let δ > 0 be a small value to be chosen later. By Egorov's Theorem ϕ ε converge uniformly to 0 on I \ Ĵ for some open set Ĵ ⊂ I with | Ĵ| < δ/2. Now consider

J = Ĵ ∪ j∈N (p j -δ/2 j+2 , p j + δ/2 j+2 )
where the points p i correspond to the support of the atomic component of θ. For the sake of clarity we may rewrite

J = ∪ i∈N C i with C i = (a i , b i ). By uniform convergence we have lim ε↓0 ϕ ε (a i ) = lim ε↓0 ϕ ε (b i ) = 1.
We set

z ε i = sup C i |ϕ ε |.
Now by Young's inequality and a change of variables we have the estimate

G ε (θ ε , ϕ ε ; (a i , b i )) ≥ b i a i [f (z ε i )|θ ε | + |ϕ ||ϕ|] dx ≥ f (z ε i ) b i a i |θ ε | dx + (z ε i ) 2 .
Applying the latter on each interval C i we get

G ε (θ ε , ϕ ε ; I) ≥ I\J f (ϕ ε )|θ ε | dx + C i f (z ε i ) C i |θ ε | dx + (z ε i ) 2
Let us pass to the liminf in the latter equation taking advantage of Fatou' lemma and the lower semicontinuity of the total variation. Since ϕ ε → 0 uniformly on I \ J and f is continuous, we have

lim inf ε↓0 G ε (θ ε , ϕ ε ; I) ≥ f (0)|θ|(I \ J) + i f (z i )|θ|(C i ) + (z i ) 2 ≥ f (0)|θ|(I \ J) + C i inf z∈(0,1) f (z i )|θ|(C i ) + (z i ) 2
Recalling the properties for f obtained in Lemma 5.2 we have

lim inf ε↓0 G ε (θ ε , ϕ ε ; I) ≥ h (0)|θ|(I \ J) + C i h (|θ|(C i )) .
We conclude observing that |θ|(C i ) ≥ |m j | if C i contains some p j and that θ coincides with θ ⊥ on I \ J therefore sending δ to zero we obtain

lim inf ε↓0 G ε (θ ε , ϕ ε ; I) ≥ h (0)|θ ⊥ |(I) + j∈N h(|m j |).
The latter lemma allows to prove the lower bound for the result in Theorem 5.1.

Γlim inf inequality for Theorem 5.1. Let (σ ε , ϕ ε ) converge to (σ, ϕ) in the considered topology. We first extend σ ε , σ, ϕ ε and ϕ to R 2 \ Ω by zero. The phasefield cost functional and the cost functional are extended to R 2 in the obvious way (their values do not change). Without loss of generality (potentially after extracting a subsequence) we may assume lim ε→0 F ε (σ ε , ϕ ε ) to exist and to be finite (else there is nothing to show). As a consequence we have div σ ε = µ + εµ - ε as well as div σ = µ +µ -and ϕ ≡ 0 (since the phasefield cost functional is bounded below by 1 2ε Ω ϕ 2 ε ). Choosing some ξ ∈ S 1 , by Fubini's decomposition theorem we have

F ε (σ ε , ϕ ε ; A) = ∞ -∞ A ξ,t f (ϕ ξ,t ε )|σ ξ,t ε | + 1 2 ε|(ϕ ξ,t ε ) | 2 + (ϕ ξ,t ε ) 2 ε δx δt = ∞ -∞ G ε (σ ξ,t ε , ϕ ξ,t ε ; A ξ,t ) dt.
Fatou's lemma thus implies

lim inf ε→0 F ε (σ ε , ϕ ε ; A) ≥ ∞ -∞ lim inf ε→0 G ε (σ ξ,t ε , ϕ ξ,t ε ; A ξ,t ) δt .
By assumption, the left-hand side is finite so that the right-hand side integrand is finite for almost all t ∈ R as well. Pick any such t and pass to a subsequence such that lim inf turns into lim. Indeed σ ξ,t ε * σ ξ,t for every ξ and almost all t, as σ ε * σ. Thus, the reduced dimension problem 5.3 implies

lim inf ε→0 G ε (σ ξ,t ε , ϕ ξ,t ε ; A ξ,t ) ≥ G (σ ξ,t , ϕ ξ,t ; A ξ,t )
for almost all t ∈ R so that

lim inf ε→0 F ε (σ ε , ϕ ε ; A) ≥ ∞ -∞ G (σ ξ,t , ϕ ξ,t ; A ξ,t ) δt .
For notational convenience let us now define the auxiliary function κ, defined for open subsets

A ⊂ R 2 , as κ(A) = lim inf ε→0 F ε (σ ε , ϕ ε ; A) .
Furthermore, introduce the nonnegative Borel measure

λ(A) = h (0)|σ ⊥ |(A) + Sσ∩A h(m σ ) dH 1
as well as the |σ|-measurable Borel functions

ψ j : R 2 → R, ψ j = σ |σ| • ξ j
for some sequence ξ j , j ∈ N, dense in S 1 . Since σ is a divergence measure vectorfield, we have

κ(A) ≥ ∞ -∞ G (σ ξ j ,t , ϕ ξ j ,t ; A ξ j ,t ) dt = ∞ -∞ h (0)|(σ ξ j ,t ) ⊥ |(A ξ j ,t ) + Sσ ξ j ,t ∩A ξ j ,t h(|m σ ξ j ,t |) dH 0 dt = h (0)|σ ⊥ • ξ j |(A) + Sσ∩A h(|m σ |)|θ σ • ξ j | dH 1 ≥ A ψ j dλ
for all j ∈ N where we have used Remark 9 in the last equality. By [Bra98, Prop. 1.16] the above inequality implies

κ(A) ≥ A sup j ψ j δλ
for any open A ⊂ R 2 . In particular, choosing A as the 1-neighborhood of Ω we obtain

lim inf ε→0 F ε (σ ε , ϕ ε ; Ω) = κ(A) ≥ A sup j ψ j δλ = h (0)|σ ⊥ |(A) + Sσ∩A h(m σ ) dH 1 = E µ + ,µ -[σ] ,
the desired result.

We now prove the associated upper bound, actually we only construct the recovery sequence for a segment. The general case can be handled as in the previous chapters of the thesis. Γlim sup inequality for Theorem 5.1. As always we concentrate on a single segment assuming σ = θH 1 Σ with Σ = [0, L] × {0} and θ = m e 1 with m > 0. Let

z m = argmin{z ∈ [0, +∞) : f (z) m + z 2 }.
For the vector field we define

σ ε = m χ {d Σ ≤ε 2 } ε 2 e 1
where d Σ is the distance function from the set Σ. For the phase-field, we let Φ be the solution of the following Cauchy problem in R

ϕ = -|ϕ|, ϕ(0) = z m ,
whose solution on [0, ∞) is given by the function Φ(t) = z e -t . Let ϕ ε be defined as

ϕ ε (x) :=    z m , if d Σ (x) ≤ ε 2 , Φ d Σ (x) -ε 2 ε , otherwise.
Chapter 5. Generalized cost functions

Considering the fact that σ ε ≡ 0 in the set {d Σ ≥ ε 2 } we have

F ε (σ ε , ϕ ε ) = {d Σ (x)≤ε 2 } f (z m ) m ε 2 dx + {d Σ (x)≥ε 2 } 1 2 |∇ϕ ε | 2 + ϕ 2 ε ε dx Let us remark that in force of the fact Ω ⊂ R 2 we have |{d Σ (x) ≥ ε 2 }| = ε 2 L + o(ε 2 ) and H 1 ({d -1 Σ (s)}) = 2 L + 2 π s.
Taking advantage of a change of variables we obtain:

F ε (σ ε , ϕ ε ) = f (z m ) m L + o(1) + ∞ 0 |Φ (t)||Φ(t)| 2 L + 2 π (εt -ε 2 ) dt.
By evaluating the integral on the righthand side directly and passing to the superior limit we conclude lim sup

ε↓0 F ε (σ ε , ϕ ε ) = (f (z m ) m + z 2 m ) L = h(m) L.
Observe that this corresponds exactly with

lim sup ε↓0 F ε (σ ε , ϕ ε ) = Σ h(m) dH 1
and the proof in the case of a segment is concluded.

Conclusion

To conclude this work we highlight some possible developments of the treated thematics. We first focus on a theoretical claim and then we will present some numerical methods which could be implemented in the future.

Let us analyze the h-mass transport problem in R 3 . In a recent work [START_REF] Burger | Second-order edgepenalization in the Ambrosio-Tortorelli functional[END_REF] the authors propose to substitute for the gradient of the phase field term in the Ambrosio-Tortorelli functional a term depending on a second order differential operator. This modification enhances the regularity of the phase fields and allows for computational and practical improvements of the existing schemes. Inspired by this idea and those of the last chapter we are led to consider a functional of the form

F ε (σ, ϕ) := Ω f (ϕ)|σ| + ε 2 |∆ϕ| 2 + ϕ 2 ε 2 dx, (5.7)
complemented with the usual divergence constraint ∇ • σ = µ +µ -. The latter functional resembles closely the one studied in the last Chapter 5 and the heuristic Γ-convergence argument is analogous. The main difference relies in the definition of the penalization function f . In order to define this function we need to consider the transition cost for the phase field which is associated to the following minimization problem

T (x) :=        min v ∞ 0 v (r) + 1 r v (r) 2 + v(r) 2 r dr, v ∈ H 2 loc ((0, +∞)), v(0) = x, v (0) = 0, lim r→+∞ v(r) = 0.
With this definition for the function T we may follow the same strategy used previously and set f = (-h * ) -1 • T . A first point of investigation would be to prove the following claim Claim 5.2. For any sequence ε ↓ 0 we have

F ε Γ - → E .
Where E is defined in equation (4.2).

This result is quite expected and the proof should follow closely the one in the Chapter 5. Indeed we could use the same Modica-Mortola component which has been used in Chapter 3 to obtain a similar result. The advantage of this choice is related to the numerical method presented below.

Conclusion

Regarding the numerical approximations it would be interesting to apply some of the techniques proposed by Bonnivard, Bretin and Lemenant in [START_REF] Bonnivard | Numerical approximation of the steiner problem in dimension 2 and 3[END_REF] to our problems. Let us recall that for fixed ε 1 the minimization problem associated to the functional defined in equation (5.5) has the form min

   Ω f (ϕ)|σ| + 1 2 ε|∇ϕ| 2 + ϕ 2 ε dx ϕ ∈ W 1,2 (Ω), σ ∈ L 1 (Ω) and ∇ • σ = µ + -µ - (5.8)
We propose an alternate minimization scheme which is suitable to the case in which either µ + or µ -is atomic. Focus on the minimization in σ for fixed ϕ, namely min

Ω f (ϕ)|σ| dx : σ ∈ L 1 (Ω) and ∇ • σ = µ + -µ -.
(5.9)

The latter is equivalent to the Beckman model for congested transportation [START_REF] Beckmann | A continuous model of transportation[END_REF] in which f (ϕ) models the congestion rate at each point. We reformulate (5.9) as a minimization problem on the set of continuous paths, namely C([0, 1], Ω). We let Γ(µ + , µ -) be the set of measures Q on C([0, 1], Ω) such that

e 0# Q = µ -and e 1# Q = µ + (5.10)
where e 0 (γ) = γ(0) and e 1 (γ) = γ(1) for any γ ∈ C([0, 1], Ω). For any [START_REF] Carlier | A continuous theory of traffic congestion and Wardrop equilibria[END_REF][START_REF] Santambrogio | A Dacorogna-Moser approach to flow decomposition and minimal flow problems[END_REF] to any such σ we may associate a measure Q ∈ Γ(µ + , µ -). Remark that

Q ∈ Γ(µ + , µ -) the expression σ = C([0,1],Ω) γH 1 γ([0, 1]) dQ defines a vector measure σ ∈ M(Ω, R n ) such that ∇ • σ = µ + -µ -, viceversa [Smi93,
Ω f (ϕ)|σ| dx = Ω f (ϕ) d|σ| = Ω f (ϕ) d C([0,1],Ω) | γ| dH 1 γ([0, 1]) dQ = C([0,1],Ω) γ([0,1]) f (ϕ) dH 1 dQ
So that the minimization problem (5.9) is equivalent to min

C([0,1],Ω) γ([0,1]) f (ϕ) dH 1 dQ : Q ∈ Γ(µ + , µ -) .
This equivalence is particularly interesting in the case µ + = δ x 0 . As a matter of fact in this case the minimizer in the above is achieved when the measure Q is supported on the geodesics, with respect to the Riemannian metric induced by f (ϕ), joining x 0 to each point in supp(µ -). Therefore the minimization procedure reduces to the problem of finding each one of these geodesics. Eventually this research can be done in a fast and efficient way by means of the Fast Marching Method [START_REF] Sethian | Fast marching methods[END_REF]. The minimization of (5.8)

Conclusion

with respect to ϕ can be done by solving via Fast Fourier Transform the associated PDE, namely ε∆ϕ -ϕ ε f (ϕ)|σ| = 0. This approach has two major benefits. Firstly, allows to minimize in the σ variable overcoming the non-differentiability of the norm, secondly it would be quite efficient since it relies on fast algorithms. Furthermore the presented method can be applied as well to the functional defined in equation (5.7). The PDE associated to the ϕ problem depends on the bilaplacian of ϕ and takes the form

ε 2 ∆ 2 ϕ - ϕ ε 2 -f (ϕ)|σ| = 0.
In this the Fast Fourier Transform would provide a better tool with respect to finite elements methods. The same method could be applied to the functional studied in Chapter 3 but would lead to a PDE with worst non-linearities. Proof.

Step 1: In order to apply the results of [START_REF] Bellettini | The Γ-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension[END_REF], we first need to modify u and the energy. Let us denote the energy density function h(t) = 1 + βt and for k ≥ 0 and t ≥ 0 let us introduce the approximation We have 0 ≤ h k ≤ h and h k ≡ h on [2 -k , +∞). Notice that h k is continuous, subadditive and increasing on [0, +∞) and that h k (0) = 0 with lim t→0 h k (t) t = +∞. We define the associated energy for functions v ∈ P C( Ω) as E h k (v, Ω) := Jv∩ Ω h k ([v]) dH 1 . Now we denote P C k ( Ω) the set of functions v ∈ P C( Ω), (1.6), such that v( Ω) ⊂ 2 -k Z. For these functions we have |v Proof. a) Let r 1 > 0 and ε > 0 such that r 1 ε ≤ r, (1 + r 1 )ε ≤ r, we define

h k (t) := (2 k/2 + β2 -k/2 ) √ t,
+ (x) -v -(x)| ≥ 2 -k for H 1 -almost every x ∈ J v . Consequently, one has E h k (v, Ω) = E h (v, Ω).
ϕ ε (x) :=      η if |x| < r 1 ε, η + (1 -η)(|x|/ε -r 1 ) if r 1 ε ≤ |x| < (1 + r 1 )ε, 1 if (1 + r 1 )ε ≤ |x| < r, and
ϑ ε (x) :=    m |B r 1 ε | if |x| < ε, 0 if ε ≤ |x| < r.
By construction, (ϕ ε , ϑ ε ) ∈ Y ε,β (m, r, r) ∩ Y ε,β (m, r). We estimate successively the three terms of the energy. First, since ε|∇ϕ ε | = (1η) ≤ 1 in B (1+r 1 )ε \ B r 1 ε and vanishes outside,

Br ε p-d |∇ϕ ε | p dx ≤ |B (1+r 1 )ε \ B r 1 ε | ε -d ≤ ω d (1 + r 1 ) d .
Next, bounding |1ϕ ε | by the characteristic function of B (1+r 1 )ε we have

Br (1 -ϕ ε ) 2 ε d dx ≤ ω d (1 + r 1 ) d .
Finally,

Br ϕ ε |ϑ ε | 2 ε dx = 1 ω d r d 1 η m 2 ε d+1 = βm 2 ω d r d 1 .
Gathering the estimates yields to the bound

max{h d ε,β (m, r, r), h d ε,β (m, r)} ≤ G ε,β (ϕ ε , ϑ ε ) ≤ 2ω d (1 + r 1 ) d + am 2 ω d r d 1 .
Then, assuming ( 1 Br ξ which is not null for ε small. We have ( θε , ϕ ε ) ∈ Y ε,β (m, r, r) if and only if r ξ ≤ r. Indeed, this holds as

C ≥ B r ξ (1 -ϕ ε ) 2 ε d dx ≥ ω d (1 -ξ) 2 r ξ ε d , (B.4)
which ensures that r ξ = O(ε). Finally let us evaluate the energy

G ε,β ( θε , ϕ ε ) = Br ε p-d |∇ϕ ε | p + (1 -ϕ ε ) 2 ε d + ϕ ε | θε | 2 ε dx = Br ε p-d |∇ϕ ε | p + (1 -ϕ ε ) 2 ε d dx + Br ξ ϕ ε m 2 |ϑ ε | 2 ε( Br ξ ϑ ε ) 2 dx ≤ m 2 ω d Br ξ ϑ ε 2 G ε,β (ϑ ε , ϕ ε ) = [1 + O(1)]G ε,β (ϑ ε , ϕ ε ).
Passing to the infimum we get

h d ε,β (m, r, r) ≤ h d ε,β (m, r) + O(1). (B.5)
Step 2: [h

d ε,β (m, r) ≤ h d ε,β (m, r, r) + o(1)] Consider a minimizing pair (ϑ ε , ϕ ε ) such that h d ε,β (m, r, r) = G ε,β (ϑ ε , ϕ ε ).
Let χ be a smooth cutoff function such that χ(x) = 1 if |x| ≤ r and χ(x

) = 0 if |x| > r+r 2 and set v ε = χϕ ε + (1 -χ). By construction (ϑ ε , v ε ) ∈ Y ε,β (m, r), furthermore, since ϕ ε ∈ (0, 1], it holds that ϕ ε ≤ v ε and (1 -ϕ ε ) 2 ≥ (1 -v ε ) 2 . Moreover as v ε ≡ ϕ ε on B r we have Br ϕ ε |ϑ ε | 2 dx = Br v ε |ϑ ε | 2 dx.
Eventually, we estimate the gradient component of the energy as follows

Br ε p-d |∇v ε | p dx = Br ε p-d |χ∇ϕ ε + (ϕ ε -1)∇χ| p dx ≤ Br ε p-d (|∇ϕ ε | + |∇χ|) p dx ≤ Br ε p-d |∇ϕ ε | p dx + C(r, χ) G 1-1/p ε,β (ϑ ε , v ε )ε p-d p + ε p-d
where we have used the inequality (|a| + |b|) p ≤ |a| p + C p (|a| p-1 |b| + |b| p ) and Holder inequality. We get

h d ε,β (m, r) ≤ G ε,β (ϑ ε , v ε ) ≤ G ε,β (ϑ ε , ϕ ε ) + O(ε p-d p ) = f r ε (m, r) + o(1) (B.6)
Step 3: Combining inequalities (B.5) and (B.6) we obtain

h d ε,β (m, r, r) -h d ε,β (m, r) = o(1).

B.2 Study of the transition energy

Given two values r 1 < r 2 let us introduce the functional

T d (v; (r 1 , r 2 )) := r 2 r 1 t d-1 |v | p + (1 -v) 2 dt
and for any triplet (ξ, r 1 , r 2 ) ∈ [0, 1] × R + × R + we set

q d (ξ, r 1 , r 2 ) := inf T d (v; (r 1 , r 2 )) : v ∈ W 1,p (r 1 , r 2 ), v(r 1 ) = ξ and v(r 2 ) = 1 . (B.7)
This value represents the cost of the transition from ξ to 1 in the ring B r 2 \ B r 1 . We will say that a function v is admissible for the triplet (ξ, r 1 , r 2 ) if it is a competitor in the above minimization problem. Let us investigate the properties of the function introduced.

B.3. Proof of Proposition 2.1

Indeed, for any λ ∈ (0, 1) the function λv 1 + (1λ)v 2 is a competitor for the minimization problem q d ∞ (λξ 1 + (1λ)ξ 2 , r), therefore it holds

q d ∞ (λξ 1 + (1 -λ)ξ 2 , r) ≤ ∞ r t d-1 |λv 1 -(1 -λ)v 2 | p + (1 -λv 1 + (1 -λ)v 2 ) 2 dt ≤ λ q d ∞ (ξ 1 , r) + (1 -λ) q d ∞ (ξ 2 , r).
Thus q d ∞ (•, r) is continuous in the open interval (0, 1). To show the continuity in 0 let ξ be small and v = argmin q d ∞ (ξ, r). Set

f (t) :=    1 1 - √ ξ (t -ξ), t < ξ, t, t ≥ ξ.
and observe that f • v is a competitor for the problem q d ∞ (0, r). Then

q d ∞ (0, r) ≤ ∞ r t d-1 |(f • v) | p + (1 -f • v 2 dt ≤ 1 (1 - √ ξ) p q d ∞ (ξ, r) + ∞ r t d-1 (1 -f • v) 2 -(1 -v) 2 dt
Let us estimate the second addend in the latter. By the definition of f we have

∞ r t d-1 (1 -f • v) 2 -(1 -v) 2 dt = {v< √ ξ} t d-1 (1 -f • v -v) 2 (v -f • v) 2 dt ≤ 4ξ {v< √ ξ} t d-1 dt ≤ 4ξ (1 - √ ξ) 2 q d ∞ (ξ, r). Since q d ∞ (•, r) is monotone we have |q d ∞ (0, r) -q d ∞ (ξ, r)| ≤ max 1 -(1 - √ ξ) p (1 - √ ξ) p , 4ξ (1 - √ ξ) 2 κ, which shows that q d ∞ (•, r) is continuous in 0.

B.3 Proof of Proposition 2.1

We show that lim inf

h d ε,β (m, r) = G ε,β (ϑ ε , ϕ ε ) ≥ Br\Br ξ ε p-d |∇ϕ ε | p + (1 -ϕ ε ) 2 ε d dx aε + Br ξ (1 -ϕ ε ) 2 ε d dx + Br ϕ ε |ϑ ε | 2 ε dx bε . (B.9)
We deal with each addend separately. First observe that by Cauchy-Schwarz inequality, it holds m 2

Br

1 ϕε dx ≤ Br ϕ ε ϑ 2 ε dx.
Plugging the latter in the term b ε of (B.9) we have

b ε ≥ Br ξ (1 -ϕ ε ) 2 ε d dx + m 2 ε Br\Br ξ 1 ϕε dx + Br ξ 1 ϕε dx taking into account η ≤ ϕ ε ≤ ξ in B r ξ , ξ ≤ ϕ ε ≤ 1 in B r \ B r ξ and η = β ε d+1 we obtain b ε ≥ ω d (1 -ξ) 2 r ξ ε d + m 2 ω d β r ξ ε d + ω d εr d ξ . (B.10) Since b ε ≤ h d ε,β (m, r) ≤ C(m)
we deduce that r ξ /ε belongs to a fixed compact subset K = K(m, ξ) of (0, +∞). Up to extracting a subsequence, which we do not relabel, we can suppose r ξ /ε to converge to some r > 0. Let us now consider the term a ε . Let v ε be the radial profile of ϕ ε

a ε = Br\Br ξ ε p-d |∇ϕ ε | p + (1 -ϕ ε ) 2 ε d dx = (d -1) ω d r/ε r ξ /ε t d-1 |v ε | p + (1 -v ε ) 2 dt.
With the notation introduced in section B.2 and Lemma B.3 therein we deduce lim inf

ε↓0 a ε ≥ (d -1) ω d lim inf ε↓0 q d (ξ; (r ξ /ε, r/ε)) ≥ (d -1) ω d q d ∞ (ξ, r),
where q d ∞ has been defined in (2.10). Combining inequality (B.10) and the latter we get

lim ε↓0 h d ε,β (m, r) ≥ (d -1) ω d q d ∞ (ξ, r) + (1 -ξ) 2 ω d rd + β m 2 ω d rd .
Sending ξ to 0 we have, by continuity (Lemma B.3) q d ∞ (ξ, r) → q d ∞ (0, r). Then taking the infimum in r, we obtain lim inf

ε↓0 h d ε,β (m, r) ≥ min r (d -1) ω d q d ∞ (0, r) + ω d rd + β m 2 ω d rd . B.4. Proof of Proposition 2.2
Again by Lemma B.3 the function q d ∞ (0, r) is nondecreasing in r, and q d ∞ (0, 0) > 0 therefore setting κ := (d -1) ω d q d ∞ (0, 0) ≤ h d β (m) we conclude the proof of Proposition 2.1.

B.4 Proof of Proposition 2.2

Let δ > 0, by Lemma B.3 for ε sufficiently small

q d (η; (r * , r/ε)) ≤ q d ∞ (0, r * ) + δ. Let v δ (t) = argmin T d v; r * , r ε dt : v (r * ) = η and v r ε = 1 .
and set

ϕ ε (t) :=    η for 0 ≤ t ≤ r * ε v δ t ε for r * ε ≤ t ≤ r
Set ϑ ε (s) to be constant equal to Let us show that the pair (ϑ ε , ϕ ε ) defined satisfy inequality (2.13). Taking advantage of the radial symmetry of the functions we get

G ε,β (ϑ ε , ϕ ε ) = r εr * t d-1 ε p+d |ϕ ε | p + (1 -ϕ ε ) ε d dt + + (1 -η) 2 ε d ω d (εr * ) d + η ε m ω d (εr * ) d 2 ω d (εr * ) d .
By simplifying the expression and considering the change of variable s = t ε in the latter it holds

G ε,β (ϑ ε , ϕ ε ) = (d -1) ω d r ε r * s d-1 [|v δ | p + (1 -v δ )] ds + (1 -η) 2 ω d r d δ + η m 2 ω d r d * ε d+1 ≤ (d -1) ω d q d (η; (r * , r/ε)) + (1 -η) 2 ω d r d * + η ε m 2 ω d r d *
Then, by Lemma B.3 for ε sufficiently small we have independently of the choices for r and r < r. For the sake of clarity we introduce

G ε,β (ϑ ε , ϕ ε ) ≤ β m 2 ω d r d B.
T (m, r) := β m 2 ω d r d + ω d r d + (d -1) ω d q d ∞ (0, r)
and recall that h d β (m) = min r T (m, r) for m > 0 and h d β (0) = 0, see (2.9).

Proof.

Let us prove the continuity of h d β on (0, +∞). For m 1 , m 2 ∈ (0, +∞) and for i = 1, 2 let r i be such that h d β (m i ) = T (m i , r i ). On one hand comparing with r = 1 it holds

m 2 i ω d-1 r d i ≤ h d β (m i ) ≤ T (m i , 1) (B.12)
on the other hand analougusly we have

ω d-1 r d i ≤ h d β (m i ) ≤ T (m i , 1). (B.13)
Consequently ω d-1 r d i belongs to the compact set [m i /T (m i , 1), T (m i , 1)]. Now remark that

h d β (m 1 ) ≤ T (m 1 , r 2 ) = h d β (m 2 ) + T (m 1 , r 2 ) -T (m 2 , r 2 ) thus |h d β (m 1 ) -h d β (m 2 )| ≤ |T (m 1 , r 2 ) -T (m 2 , r 2 )| ≤ |m 2 1 -m 2 2 | ω d-1 min{r d 1 , r d 2 }
and taking into account inequality (B.12) we have

|h d β (m 1 ) -h d β (m 2 )| ≤ (m 1 + m 2 ) max T (m 1 , 1) m 2 1 , T (m 2 , 1) m 2 2 |m 1 -m 2 |.
Observing that T (•, 1) is continuous we conclude that h d β is continuous on (0, +∞). Next, we see that

h d β is non decreasing. Let 0 < m 1 < m 2 and r > 0. Let (ϑ, ϕ) ∈ Y ε,β (m 2 , r) such that G ε,β (ϑ, ϕ; B r ) = h d ε,β (m 2 , r) . Set ϑ = m 1 ϑ/m 2
and remark that the pair (ϑ, u) belongs to Y ε,β (m 1 , r). Therefore we have the following set of inequalities

h d ε,β (m 1 , r) ≤ G ε,β (ϑ, ϕ; B r ) = G ε,β m 1 ϑ m 2 , ϕ; B r < G ε,β (ϑ, ϕ; B r ) = h d ε,β (m 2 , r).
Passing to the limit as ε ↓ 0 we obtain

h d β (m 1 ) ≤ h d β (m 2 ).
supported function in C 0 (R n ) or C 1 (R n ) can be arbitrarily well approximated (in the respective norm) by finite linear combinations of tensor products (φ • π H )(ψ • π ξ ) with φ ∈ C ∞ (H) and ψ ∈ C ∞ (R) with compact support. Thus, the above implies R H ξ,s θ(y) dν ξ,s (y) dσ ⊥ ξ (s) = 0 for any compactly supported θ ∈ C 0 (R n ) so that ν ξ,s ⊗ σ ⊥ ξ (s) = 0 and thus σ ⊥ ξ (s) = 0 .

Summarizing, we have σ • ξ = σ ξ (s)ν ξ,s ⊗ ds and

H ξ,s φ(π H (x))σ ξ (s) dν ξ,s (x) ds = {ξ•x<s} ∇[φ • π H ](x) • dσ(x) + {ξ•x<s} φ(π H (x)) d div σ(x)
for all compactly supported φ ∈ C ∞ (H). Note that the right-hand side is left-continuous in s so that the left-hand side is as well. Consequently, σ ξ (s)ν ξ,s is left-continuous in s with respect to weak-* convergence. Now let χ ∈ C ∞ (R) with χ = 1 on (-∞, 0], χ = 0 on [1, ∞), and 0 ≤ χ ≤ 1, and define for ρ > 0

χ ρ (x) = χ π ξ (x)-t ρ , σ ρ = χ ρ σ, µ ρ = χ ρ div σ, σ ρ ξ,s = 1 ρ χ π ξ (•)-t ρ σ ξ (s)ν ξ,s .
In the distributional sense we have div σ ρ = µ ρ + σ ρ ξ,s ⊗ ds so that for any compactly supported θ ∈ C ∞ (R n ) we have

R n ∇θ • dσ ρ + R n θ dµ ρ = - R H ξ,s
θ dσ ρ ξ,s ds .

Letting ρ → 0 and using the left-continuity of σ ξ (s)ν ξ,s in s we arrive at (C.1).

We now define the slice of a divergence measure vector field as the measure obtained via disintegration with respect to the one-dimensional Lebesgue measure.

Definition 3 (Sliced sets, functions, and measures). Let ξ ∈ S n-1 and t ∈ R.

For

A ⊂ R n we define the sliced set A ξ,t = A ∩ H ξ,t . 2. For f : A → R we define the sliced function f ξ,t : A ξ,t → R, f ξ,t = f | A ξ,t . For f : A → R n we define f ξ,t : A ξ,t → R n , f ξ,t = ξ • f | A ξ,t .
3. We define the sliced measure of a compactly supported divergence measure vector field σ as σ ξ,t = σ ξ (t) ν ξ,t .

By Lemma C.1 it holds σ • ξ = σ ξ,t ⊗ dt.
Remark 6 (Properties of sliced functions and measures). 1. By Fubini's theorem it follows that for any function f of Sobolev-type W m,p the corresponding sliced function f ξ,t is well-defined and also of Sobolev-type W m,p for almost all ξ ∈ S n-1 and t ∈ R. For the same reason, strong convergence f j → j→∞ f in W m,p implies strong convergence (f j ) ξ,t → f ξ,t in W m,p on the sliced domain.

2. The definitions of sliced functions and measures are consistent in the following sense. If we identify a Lebesgue function f with the measure χ = f L for L the Lebesgue measure, then the same identification holds between f ξ,t and χ ξ,t for almost all ξ ∈ S n-1 and t ∈ R.

3. Let σ be a divergence measure vector field, then the properties [AFP00, Thm. 2.28] of the disintegration σ

• ξ = ν ξ,t ⊗ π ξ # |σ • ξ|(t) = ν ξ,t ⊗ σ ξ (t) dt = σ ξ,t ⊗ dt im- mediately imply the following. The map t → σ ξ,t M is integrable and satisfies R σ ξ,t M dt = R σ ξ (t) dt = σ • ξ M .
Furthermore, for any measurable function f : R n → R, absolutely integrable with respect to |σ • ξ|, it holds

R n f (x) dσ•ξ = R H ξ,t f (x) dν ξ,t (x) dπ ξ # |σ•ξ|(t) = R H ξ,t f (x) dσ ξ,t (x) dt .
We briefly relate our definition of sliced measures to other notions of slices from the literature.

Remark 7 (Notions of slices).

1. Let Lip(A) denote the set of bounded Lipschitz functions on A ⊂ R n . An alternative definition of the slice of a divergence measure vector field σ was introduced by Šilhavý [ Š07] as the linear operator

σ ξ,t : Lip(H ξ,t ) → R, σ ξ,t (ϕ| H ξ,t ) = lim δ 0 1 δ {x∈R n | t-δ<x•ξ<t} ϕξ • dσ (C.2)
for all ϕ ∈ Lip(R n ) (the right-hand side is well-defined and only depends on ϕ| H ξ,t [ Š07, Thm. 3.5 & Thm. 3.6]). This σ ξ,t equals the so-called normal trace of σ on H ξ,t (see [ Š07] for its definition and properties). In general it is not a measure but continuous on Lip(H ξ,t ) in the sense

σ ξ,t (ϕ) ≤ ( σ M + div σ M ) ϕ W 1,∞ for all ϕ ∈ Lip(H ξ,t ) .
2. Interpreting a divergence measure vector field as a 1-current or a flat 1-chain, Šilhavý's definition of σ ξ,t is identical to the classical slice of σ on H ξ,t as for instance defined in [START_REF] White | Rectifiability of flat chains[END_REF] or [START_REF] Federer | Geometric measure theory[END_REF]4.3.1] (note that Šilhavý's definition corresponds to [ Š07, (3.8)], whose analogue for currents is [Fed69, 4.3.2(5)]).

3. Our notion of a sliced measure from Definition 3 is equivalent to both abovementioned notions. Indeed, (C.1) implies

σ ξ,t = (div σ) {x • ξ < t} -div(σ {x • ξ < t}) ,
which shows that the sliced measure represents the normal flux through the hyperplane H ξ,t = {x • ξ = t}. This, however, is the same characterization as given in [ Š07, (3.6)] and [Fed69, 4.2.1] for both above notions of slices.

We conclude the section with several properties needed for the Γ-lim inf inequality. The following result makes use of the Kantorovich-Rubinstein norm (see for instance [LLSV14, eq. ( 2) & (5)]; in geometric measure theory it is known as the flat norm) on M(R n ), defined by

µ KR = inf{ µ 1 M + µ 2 M | µ 1 ∈ M(R n ), µ 2 ∈ M(R n ; R n ), µ = µ 1 + div µ 2 } = sup Ω f dµ f Lipschitz with constant 1, |f | ≤ 1 .
For measures of uniformly bounded support and uniformly bounded mass it is known to metrize weak- * convergence (see for instance [BW17, Rem. 2.29(3)-(4)]). We will furthermore make use of the following fact. Let T s : x → xsξ be the translation by s in direction -ξ. It is straightforward to check that for any divergence measure vector field

µ ∈ M(R n ; R n ) we have div(π H ξ,t # (µ -µ • ξ ξ)) = π H ξ,t # ( div µ) .
As a consequence, for any µ ∈ M(H ξ,t ) and ν ∈ M(H ξ,t+s ) we have

µ -ν KR ≥ µ -T s# ν KR . Indeed, let δ > 0 arbitrary and µ 1 ∈ M(R n ), µ 2 ∈ M(R n ; R n ) with µ-ν = µ 1 +div µ 2 such that µ -ν KR ≥ µ 1 M + µ 2 M -δ, then μ1 = π H ξ,t # µ 1 and μ2 = π H ξ,t # (µ 2 - µ 2 • ξ ξ) satisfy µ -T s# ν = μ1 + div μ2 and thus µ -T s# ν KR ≤ μ1 M + μ2 M ≤ µ 1 M + µ 2 M ≤ µ -ν KR + δ .
Theorem C.1 (Weak convergence of sliced measures). Let σ j * σ as j → ∞ for a sequence {σ j } of compactly supported divergence measure vector fields with uniformly bounded div σ j M . Then for almost all ξ ∈ S n-1 and t ∈ R we have

σ j ξ,t * σ ξ,t .
Proof. It suffices to show σ j ξ,t * σ ξ,t for a subsequence.

Consider the measures ν j = |σ j | + | div σ j |. Since ν j M is uniformly bounded, a subsequence converges weakly- * to some compactly supported nonnegative ν ∈ M(R n ) (the subsequence is still indexed by j). For I ⊂ R introduce the notation H ξ,I = t∈I H ξ,t . Then for almost all t ∈ R, ν(H ξ,[t-s,t+s] ) → 0 as well as (|σ| + | div σ|)(H ξ,[t-s,t+s] ) → 0 as s 0. For such a t we show convergence of σ j ξ,tσ ξ,t to zero in the Kantorovich-Rubinstein norm which implies weak- * convergence. To this end fix some arbitrary δ > 0. Given ζ > 0 let ρ ζ = ρ(•/ζ)/ζ for a nonnegative smoothing kernel ρ ∈ C ∞ (R) with support in [-1, 1] and R ρ dt = 1. For any compactly supported divergence measure vector field λ we now define the convolved slice λ ξ,ζ,t by

H ξ,t g dλ ξ,ζ,t = R ρ ζ (-s) H ξ,t g dT s# λ ξ,t+s ds = R ρ ζ (-s) H ξ,t g • T s dλ ξ,t+s ds ∀g ∈ C(H ξ,t ) ,
where T s : x → xsξ is the translation by s in direction -ξ. By Remark 6(3) we have σ j ξ,ζ,t * σ ξ,ζ,t . Furthermore, there exist ζ > 0 and J ∈ N such that σ ξ,tσ ξ,ζ,t KR ≤ δ 3 and σ j ξ,tσ j ξ,ζ,t KR ≤ δ 3 for all j ≥ J. Indeed, for a compactly supported divergence measure vector field λ we have

λ ξ,t -λ ξ,ζ,t KR ≤ R ρ ζ (-s) λ ξ,t -T s# λ ξ,t+s KR ds ≤ R ρ ζ (-s) λ ξ,t -λ ξ,t+s KR ds = R ρ ζ (-s) div(λ H ξ,[t,t+s) ) -(div λ) H ξ,[t,t+s) KR ds ≤ R ρ ζ (-s) |λ|(H ξ,[t,t+s) ) + | div λ|(H ξ,[t,t+s) ) ds ≤ |λ|(H ξ,[t-ζ,t+ζ] ) + | div λ|(H ξ,[t-ζ,t+ζ] ) ,
where in the equality we employed Remark 7(3). Thus, we can simply pick

ζ such that |σ|(H ξ,[t-ζ,t+ζ] )+| div σ|(H ξ,[t-ζ,t+ζ] ) ≤ δ 3 and ν(H ξ,[t-ζ,t+ζ] ) ≤ δ 6 , while we choose J such that (ν j -ν)(H ξ,[t-ζ,t+ζ] ) ≤ δ
6 for all j > J. Now let J ≥ J such that σ j ξ,ζ,t -σ ξ,ζ,t KR ≤ δ 3 for all j ≥ J, then we obtain

σ j ξ,t -σ ξ,t KR ≤ σ j ξ,t -σ j ξ,ζ,t KR + σ j ξ,ζ,t -σ ξ,ζ,t KR + σ ξ,ζ,t -σ ξ,t KR ≤ δ
for all j > J. The arbitrariness of δ concludes the proof.

Remark 8 (Flat convergence of sliced currents). The convergence from Theorem C.1 is consistent with the following property of slices of 1-currents: If σ j , j ∈ N, is a sequence of 1-currents of finite mass with σ j → σ in the flat norm, then (potentially after choosing a subsequence) σ j ξ,t → σ ξ,t in the flat norm for almost every ξ ∈ S n-1 , t ∈ R (see [CDRMS17, step 2 in proof of Prop. 2.5] or [Whi99b, Sec. 3]).

Remark 9 (Characterization of sliced measures).

1. Let the compactly supported divergence measure vector field σ be countably 1-rectifiable, that is, σ = θmH 1 S for a countably 1-rectifiable set S ⊂ R n and H 1 S-measurable functions m : S → [0, ∞) and θ : S → S n-1 , tangent to S H 1 -almost everywhere. Then the coarea formula for rectifiable sets [START_REF] Federer | Geometric measure theory[END_REF]Thm. 3

.2.22] implies |θ • ξ|H 1 S = H 0 S ξ,t ⊗ H 1 (t) so that R n f dσ • ξ = S f mθ • ξ dH 1 = R S ξ,t f m sgn(ξ • θ) dH 0 dt
for any Borel function f . Hence, for almost all t, σ ξ,t = sgn(ξ • θ) mH 0 S ξ,t .

The choice f = τ (m) m sgn(ξ • θ) yields S τ (m)|θ • ξ| dH 1 = R S ξ,t τ (m) dH 0 dt .
2. Let the compactly supported divergence measure vector field σ be H 1 -diffuse, that is, it is singular with respect to the one-dimensional Hausdorff measure on any countably 1-rectifiable set. Then for almost all ξ ∈ S n-1 and t ∈ R, σ ξ,t is H 0 -diffuse, that is, it does not contain any atoms. Indeed, let σ ξ,t have an atom at x ∈ H ξ,t , then

x ∈ Θ(σ) = x ∈ R n lim inf ρ 0 |σ|(B ρ (x))/ρ > 0 ,
where B ρ (x) denotes the open ball of radius ρ centred at x. This can be deduced as follows. Let φ ∈ C ∞ (R) be smooth and even with support in (-1, 1) and φ(0) = sgn(σ ξ,t ({x})). Further abbreviate

K = max x∈R |φ (x)| > 0 and φ ρ = φ(| • -x|/ρ) for any ρ > 0. Equation (C.1) now implies H ξ,t φ ρ dσ ξ,t = {ξ•x<t} φ ρ d div σ + {ξ•x<t} ∇φ ρ • dσ ≤ {ξ•x<t} φ ρ d div σ + K |σ|(B ρ (x)) ρ .
Taking on both sides the limit inferior as ρ → 0 we obtain

|σ ξ,t |({x}) ≤ K lim inf ρ 0 |σ|(B ρ (x))/ρ,
as desired. As a result, for a given ξ the set of t such that σ ξ,t is not H 0 -diffuse is a subset of π ξ (Θ). Thus it remains to show that for almost all ξ ∈ S n-1 the set π ξ (Θ) is a Lebesgue-nullset. Writing

Θ = p∈N Θ p for Θ p = x ∈ R n lim inf ρ 0 σ(B ρ (x))/ρ ≥ 1 p ,
it actually suffices to show that π ξ (Θ) is a Lebesgue-nullset for any p ∈ N. Now by the properties of the 1-dimensional density of a measure [AFP00, Thm. 256],

H 1 (Θ p ) ≤ p 2 |σ|(Θ p )
so that Θ p can be decomposed into a countably 1-rectifiable and a purely 1unrectifiable set [AFP00, p. 83], Le but de cette thèse est de concevoir des approximations de certains problèmes de Transport Branché. Le transport branché est un cadre mathématique de modélisation des réseaux de distribution offre-demande qui est plus général que le problème Steiner présenté ci-dessus. En particulier les usines d'approvisionnement et les lieux de demande sont modélisés comme des mesures supportées sur des points et le réseau est interprété comme une mesure vectorielle, enfin le problème est présenté comme un problème d'optimisation sous contraintes. Le coût de transport d'une masse m le long d'un bord de longueur est h(m) et le coût total d'un réseau est défini comme la somme de la contribution sur tous ses bords. Le cas de transport branché consiste dans le choix spécifique h(m) = |m| α avec α ∈ [0, 1). La sous-additivité de la fonction

Θ p = Θ r p ∪ Θ u p (Θ u p purely 1-unrectifiable means H 1 (Θ u p ∩ f (R)) = 0 for any Lipschitz f : R → R n ).
(1, 0) (-1/2, √ 3/2) (-1/2, - √ 3/2) (1, 0) (-1/2, √ 3/2) (-1/2, - √ 3/2) Figure D
.3: On the left: Minimal Spanning Tree connecting three points situated at the vertices of an equilateral triangle (longueur = 2 √ 3). Sur la droite: Steiner Minimal Tree contraint de connecter le même ensemble de points (longueur = 3). En bleu foncé le sommet supplémentaire qui permet de diminuer la longueur totale. de coût, h(m 1 + m 2 ) ≤ h(m 1 ) + h(m 2 ), assure que transporter deux masses conjointement est moins cher que de le faire séparément. Cette formulation partage la plupart des complexités numériques présentées ci-dessus dans le cas du problème de l'arbre minimal de Steiner. Dans ce travail, nous introduisons diverses approximations variationnelles au moyen de fonctions de type elliptique pour obtenir des schémas numériques plus efficaces. Finalement, la méthode proposée est généralisée aux problèmes de type Plateau qui est un cadre pour modéliser les films de savon couvrant une frontière donnée. Dans sa formulation plus générale, l'inconnu de ces problèmes est une surface k-dimensionnelle en R n enjambant une frontière (k -1)-dimensionnelle et minimisant un certain coût. Le transport branché correspond à un problème de type Plateau pour le choix k = 1.

Description du modèle

= i m i τ i H 1 Σ i . L'action de σ sur C 0 (Ω, R n ) est définie par la formule suivante (σ, ϕ) = i Σ i m i ϕ • τ i dH 1 pour tout ϕ ∈ C 0 (Ω, R n ).
Une fonction de coût de transport h : RR → [0, +∞) est une application telle que h est pair, semi-continu inférieurements, sous-additif, avec h(0) = 0. (D.3) Étant donné une fonction de coût de transport h, nous définissons le énergie de Gilbert sur la mesure vectorielle polyédrique comme suit

E h (σ) := i h(m i )H 1 (Σ i ).
Nous dotons M(Ω, R n ) avec sa topologie faible- * et étendons E h sur cet espace par relaxation, à savoir pour une mesure vectorielle σ nous fixons Pour modéliser la source et le puits du réseau de transport, nous introduisons deux mesures de probabilité µ + , µ -∈ P(Ω) et limitons notre attention à l'espace vectoriel

E h (σ) := inf lim inf j→+∞ E h (σ j ) : σ j * σ et σ j polyhedral . (
X µ + ,µ -⊂ M(Ω, R n ) composé de ces mesures vectorielles σ satisfaisant div σ = µ + -µ - (D.5)
dans le sens de distributions. Comme est montre dans la note [START_REF] Chambolle | Strong approximation in h-mass of rectifiable currents under homological constraint[END_REF] si la relaxation est obtenu par rapport aux mesures polyédriques en X µ + ,µ -nous obtenons toujours le fonctionnel (D.4).

Enfin, nous sommes intéressés à approcher les minimiseurs de l'énergie de Gilbert sous la contrainte de divergence (D.5), à savoir:

min {E h (σ) : σ ∈ X µ + ,µ -} . (D.
m h(m) h(x) = |x| 2) m h(m) h(m) = |m| α 3) m h(m) h(m) = χ {m =0} 4) m h(m) h(m) = (1 + β |m|)χ {m =0} 5) m h(m) h(m) = min{α 0 |m|, α 1 |m| + β 1 } Figure D
.5: Pour h comme dans les graphes nous obtenons respectivement le : 1) Masse, 2) α-Masse, 3) Taille, 4) Coût affine, 5) Planification urbaine fonctionnelle.

Theorem D.1 (Théorème de l'existence). Donné α ∈ (1 -1 n , 1] et deux mesures de probabilité µ + , µ -∈ P(Ω), il existe une mesure a valeurs vectorielle σ ∈ X µ + ,µ -pour laquelle E h (σ) est minimal. De plus, nous avons l'estimation suivante

E h (σ) ≤ 1 2 1-n(1-α)-1 √ n diam(Ω) 2 .
Dans un résultat subséquent [Xia04

∈ P(Ω), σ ∈ M(Ω, R n ) à variation totale finie et telle que div σ = µ + -µ -alors σ peut être décomposé en tant que σ = σ ⊥ + m τ H 1 Σ où (m, τ, Σ) est le composant H 1 -rectifiable de σ et σ ⊥ est le composant diffus. De plus E h (σ) = h (0)|σ ⊥ | + Σ h(m) dH 1 . (D.9)
Lorsqu'avec un abus de notation, nous avons dénoté h (0) = lim m↓0 h(m)/m.

Avant d'introduire des problèmes impliquant des surfaces et d'autres objets de dimensions supérieures, soulignons le fait que le problème de l'arbre minimal Steiner reliant certains points {x 0 , . . . , x N } peut être modélisé dans le contexte du transport Branché. Tout d'abord, avec le choix α = 0, E h se réduit à le fonctionnelle de taille. Deuxièmement, la contrainte de divergence oblige toute mesure vectorielle considérée à joindre le support de µ + au support de µ -donc, en choisissant µ + = δ x 0 et µ -= 1/N N i=1 δ x i nous forçons x 0 à être connecté à chaque x i . En rassemblant tous ensemble, avec ces choix, un minimiseur σ of (D.6) est supporté sur un ensemble reliant chaque couple de points dans {x 0 , . . . , x N } et a un support avec une longueur totale minimale donc est une solution du problème (D.1). L'énergie introduite ci-dessus pour les mesures rectifiables supportées sur des surfaces 1-dimensionnelles peut être généralisée à n'importe quelle dimension k ∈ {1, . . . , n}. Pour ce faire, il est nécessaire d'introduire le concept de k-courants dans R n . Dénotez avec D k (Ω) l'espace des formes différentielles lisses sur l'ensemble ouvert Ω. L'espace vectoriel de k-courants, D k (Ω), est le dual de D k (Ω) et il est naturellement doté de sa topologie faible- * . Nous suivons principalement la notation de [START_REF] Krantz | Geometric integration theory. Cornerstones[END_REF][START_REF] Federer | Geometric measure theory[END_REF] la principale différence étant l'utilisation de σ pour désigner un k-courant au lieu d'une lettre alphabétique majuscule. Pour un courant on peut définir une notion de frontière par dualité comme suit ∂σ, ω = σ, dω pour tous les (k -1) formes différentiels ω.

Nous appelons masse d'un k-courant la borne supérieure σ, ω parmi toutes les formes différentielles k avec comass délimité par 1, et on le désigne avec |σ|. En particulier par le théorème de Radon-Nikodym, nous pouvons identifier un k-courant σ à masse finie avec la mesure à valeurs vectorielle τ µ σ où µ σ est une mesure finie et positive, et τ est une fonction µ σ -mesurable a valeurs dans l'ensemble de k-vecteurs unitaires pour la norme de masse. La relation avec la mesure vectorielle est évidente quand on considère le fait que les espaces vectoriels Λ 1 R n , Λ 1 R n s'identifient avec R n . Ainsi, chaque measure vectorielles σ ∈ M(Ω, R n ) de masse finie s'identifie avec un 1-courant de masse finie et vice-versa. De plus, l'opérateur de divergence agissant sur les mesures au sens de distribution est défini par la dualité comme l'opérateur de frontière pour les courants. Donc, en analogie avec ce qui a été présenté pour les mesures vectorielles, La séquence xε est appelée recovery sequence pour x. La condition (UB) est souvent difficile à prouver donc il est pratique de trouver un sous-ensemble D ⊂ X tel que : pour chaque x ∈ X il existe une séquence approchante (x n ) ⊂ X tel que x n → x et F (x n ) → F (x). Si nous sommes capables de retrouver D alors un simple argument diagonal montre qu'il suffit de vérifier la condition (UB) pour tous les x ∈ D plutôt que pour chaque x ∈ X. Dans le contexte de notre travail, l'ensemble D correspond à l'espace vectoriel des mesures vectorielles polyédriques. Puisque la définition de Γconvergence peut paraître encombrante, nous donnons une caractérisation alternative qui permet de goûter sa pertinence dans le contexte du Calcul des Variations. Notre stratégie est de remplacer l'énergie singulière E h par une séquence de fonctionnels de type elliptique plus lisse F ε et de prouver que F ε Γ -→ E h . Puis nous prouvons que la famille (F ε ) est equicoercive : toute séquence de minima (x j ) est précompacte en X. Ceci assure que la séquence de minimiseurs xε convergent vers un minimum. Enfin, nous cherchons des méthodes numériques pour approcher un minimum xε .

On va exposer trois exemples remarquables de convergence Γ. Considérons un conteneur Ω ⊂ R 3 de volume unitaire contenant deux liquides non miscibles modélisés par une fonction binaire ϕ : Ω → Ω → {0, 1} de sorte que Ω |ϕ| dx = V ∈ (0, 1) représente le pourcentage d'un liquide par rapport au volume du récipient. On associe au système une énergie en fonction de la tension superficielle, en supposant qu'elle est directement proportionnelle à la surface de l'interface J ϕ entre les liquides. M (ϕ) = cH 2 (J ϕ ).

Une autre façon d'étudier les systèmes est de supposer que la transition n'est pas donnée par une interface de séparation infinitésimale, mais qu'il s'agit plutôt d'un phénomène continu se produisant dans une fine épaisseur de taille ε. Compte tenu de cela, Cahn et Hilliard [START_REF] Cahn | Free energy of a nonuniform system. i. interfacial free energy[END_REF] envisagent une fonction de phase continue ϕ : Ω → [0, 1] représentant le mélange ponctuel entre les fluides et supposons une énergie du type

Ω ε 2 |∇ϕ| 2 + ϕ 2 (1 -ϕ) 2 dx.
Le terme ϕ 2 (1ϕ) 2 est appelé un potentiel double puits et pénalise les valeurs loin de 0 ou 1 et l'inhomogénéité n'est pas favorisée par le terme de gradient. Le lien entre les deux énergies a été découvert par Modica et Mortola dans leurs papiers [START_REF] Modica | Il limite nella Γ-convergenza di una famiglia di funzionali ellittici[END_REF][START_REF] Modica | Un esempio di Γ --convergenza[END_REF]. Leur résultat est plus général, ils prouvent qu'un rescaling approprié de l'énergie ci-dessus Γ converge vers le périmètre fonctionnel pour toutes le dimension de domaine.

Pour approcher l'énergie S , Ambrosio et Tortorelli ont proposé la famille des fonctionnelles

S ε (u, ϕ) = Ω |∇u| 2 ϕ + β 4 ε|∇ϕ| 2 + (1 -ϕ) 2 ε dx + α Ω (u -g) 2 dx.
Dans les articles [AT90, AT92] il est prouvé que S Γ -→ S . Donnons une idée heuristique derrière ce résultat. Puisque u est proche de g dans le cas d'une discontinuité forte de g, le terme de gradient |∇u| explose. En effet, valeurs élevées dans le gradient |∇u| sont contrôlées par des valeurs proches de zéro dans la fonction d'état ϕ. D'autre part, le terme entre crochets pénalise fortement les valeurs de ϕ loin de 1. La concurrence des termes en ϕ se traduit par le fait que 1-ϕ représente une version lissée de la fonction 1χ Ju . Enfin dans la limite ε ↓ 0 le terme Modica-Mortola converge vers la taille H n-1 de l'ensemble {ϕ = 1} qui contient l'ensemble de saut de u. Les fonctionnelles modelées sur celles d'Ambrosio et de Tortorelli et cette dernière fonctionnelle elle-même sont souvent connues sous le nom d'approximations de champ de phase. Ce n'est pas seulement à cause de la relation stricte avec la fonctionnelle de Modica-Mortola mais aussi parce que nous pouvons interpréter la fonction ϕ comme une fonction d'état, elle acquiert la valeur 0 sur l'ensemble de saut de u, c'est-à-dire sur l'ensemble de discontinuité forte de la fonction, et la valeur 1 oú u est suffisamment lisse. Les deux comportements de u sont alors interprétés comme deux états possibles et ϕ modélise la fonction d'état dans chaque point pour le système. Cette observation a été prise en considération dans les travaux sur la théorie de la fracture d'Iurlano et al. [START_REF] Conti | Phase field approximation of cohesive fracture models[END_REF][START_REF] Iurlano | Fracture and plastic models as Γ-limits of damage models under different regimes[END_REF]. Lá, ϕ modélise l'état de détérioration d'un matériau et u est remplacé par une fonction de déplacement.

Pour conclure la section, nous présentons une variation de la fonctionnelle de Ambrosio-Tortorelli proposée par Bonnivard, Lemenant et Santambrogio [START_REF] Lemenant | A Modica-Mortola approximation for the Steiner problem[END_REF][START_REF] Bonnivard | Approximation of length minimization problems among compact connected sets[END_REF] pour récupérer dans la limite la fonctionnelle associée au problème de l'arbre minimal Steiner pour certains points {x 0 , . . . , x N } ⊂ Ω ⊂ R 2 . Étant donné une fonction continue ϕ : Ω → [0, 1] les auteurs introduisent une distance géodésique dépendant de ϕ, à savoir Dans les Quatrième et Cinquième Chapitres, nous limitons à nouveau notre attention aux ensembles Ω ⊂ R 2 et développons deux fonctions pour l'approximation de toutes fonctions de coàt de transport h concave et continue. Remarquez que nous disons qu'une fonction de coàt de transport est concave si c'est une fonction paire dont la restriction à [0, +∞) est concave. Le premier résultat concerne les fonctions de coàt de transport h de la forme h(m) = min{α i |m| + β i : 0 ≤ i ≤ N }. pour α 0 > α 1 > . . . > α N ≥ 0 et 0 ≤ β 0 < β 1 < . . . < β N . Notre approche profite du résultat du premier chapitre dans lequel nous avons récupéré dans la Γ-limite des fonctions de coàt de transport affine de la forme 1 + β|m|. Dans le cas N = 1 et β 0 > 0 l'énergie de champ de phase proposée prend la forme suivante

Ω min ϕ 2 0 + α 2 0 ε 2 β 0 ; ϕ 2 1 + α 2 1 ε 2 β 1 |σ| 2 2ε dx + β 0 T ε (ϕ 1 ) + β 1 T ε (ϕ 2 )
oú T ε est une énergie du type Modica-Mortola définie comme suit

T ε (ϕ) = 1 2 Ω ε|∇ϕ(x)| 2 + (ϕ(x) -1) 2 ε dx.
Soulignons la présence de deux champs de phase qui interagissent dans la composante contrainte de la fonctionnelle. Idéalement, chaque 1ϕ i est une fonction d'indicateur lisse d'un sous-ensemble du support de la mesure rectifiable limite σ. En particulier ϕ i = 0 si le choix de la composante i-th dans la définition de h est optimal par rapport à l'intensité du flux de σ. Tout le Quatrième Chapitre est consacré à la preuve du résultat de Γ-convergence et à l'étude des méthodes numériques développées en collaboration avec Carolin Rossmanith et Benedikt Wirth de l'Université de Munster.

Dans le dernier chapitre de la thèse, nous étudions des fonctionnelles de la forme Ensuite, en choisissant f (t) = (-h * ) -1 • c W (t) le màme résultat de Γ-convergence peut àtre obtenu avec une famille de fonctionnels définie comme suit

F ε (σ, ϕ) := Ω f (ϕ)|σ| + 1 2 ε|∇ϕ| 2 + W (ϕ) ε dx.
En vigueur de ce degré de liberté dans le choix du potentiel W nous commençons à analyser quel serait le meilleur choix. Ces questions, ainsi que d'autres, sont le sujet de la section finale qui examine les développements possibles des méthodes proposées. Abstract : In this thesis we devise phase field approximations of some Branched Transportation problems. Branched Transportation is a mathematical framework for modeling supply-demand distribution networks which exhibit tree like structures. In particular the network, the supply factories and the demand location are modeled as measures and the problem is cast as a constrained optimization problem. The transport cost of a mass m along an edge with length is h(m) and the total cost of a network is defined as the sum of the contribution on all its edges. The branched transportation case consists with the specific choice h(m) = |m| α where α is a value in [0, 1). The sub-additivity of the cost function ensures that transporting two masses jointly is cheaper than doing it separately. In this work we introduce various variational approximations of the branched transport optimization problem. The approximating functionals are based on a phase field representation of the network and are smoother than the original problem which al-lows for efficient numerical optimization methods. We introduce a family of functionals inspired by the Ambrosio and Tortorelli one to model an affine transport cost functions. This approach is firstly used to study the problem any affine cost function h in the ambient space R 2 . For this case we produce a full Γconvergence result and correlate it with an alternate minimization procedure to obtain numerical approximations of the minimizers. We then generalize this approach to any ambient space and obtain a full Γconvergence result in the case of k-dimensional surfaces. In particular, we obtain a variational approximation of the Plateau problem in any dimension and co-dimension. In the last part of the thesis we propose two models for general concave cost functions. In the first one we introduce a multiphase field approach and recover any piecewise affine cost function. Finally we propose and study a family of functionals allowing to recover in the limit any concave cost function h.

Universit é Paris-Saclay

Espace Technologique / Immeuble Discovery Route de l'Orme aux Merisiers RD 128 / 91190 Saint-Aubin, France

Figure 4 :

 4 Figure 4: Example of a surface spanning a 1-dimensional boundary consisting of three oriented circles.

  m) dH k + δ and |σ| ≤ |σ| + δ. Introduction In addition Colombo et al. in [CDRMS17, Proposition 2.7] have shown that the condition lim m↓0 h(m) m = +∞ is equivalent to the fact E h (σ) finite if and only if σ is rectifiable.

Figure 6 :

 6 Figure 6: Computed mass flux σ and phase fields ϕ 1 , ϕ 2 , ϕ 3 for the cost function shown on the right, ε = 0.005. The color in σ indicates which phase field is active. The result is obtained by optimizing the functional defined in Chapter Three.

1. 2 .

 2 Preliminaries for the chapter 1.2 Preliminaries for the chapter In the following Ω ⊂⊂ Ω ⊂ R d are bounded open convex sets. Given X ⊂ R d (in practice X = Ω or X = Ω), we denote by A(X) the class of all relatively open subsets of X and by A S (X) the subclass of all simply connected relatively open sets O ⊂ X such that O ∩ S = ∅. We denote by (e 1 , . . . , e d ) the canonical orthonormal basis of R d , by | • | the euclidean norm and by •, • the euclidean scalar product in R d . The open ball of radius r centered at x ∈ R d is denoted by B r (x). The (d -1)-dimensional Hausdorff measure in R d is denoted by H d-1 . We write |E| to denote the Lebesgue measure of a measurable set E ⊂ R d . When µ is a Borel meaure and E ⊂ R d is a Borel set, we denote by µ E the measure defined as µ E(F ) = µ(E ∩ F ). Let us remark that from Section 1.4 onwards, we work in dimension d = 2. For any fixed couple (σ, ϕ), with F ε (σ, ϕ; O) we denote the value of the functional (1.2) on any set O ∈ A(Ω). Similarly we define the with version E β (σ, ϕ; O) the localization of E β to O.

  and let C ε = {I connected component of B ε : I ∩ A ε = ∅}. (1.12) Let us show that the cardinality of C ε is bounded by a constant independent of ε. Let ε be fixed and consider an interval I ∈ C ε . Let a, b ∈ Ī such that {ϕ ε (a), ϕ ε (b)} = {1/2, 3/4}. Using the usual Modica-Mortola trick, we have

Figure 1 . 1 :

 11 Figure 1.1: Example of the construction of the H 1 -rectifiable measure γ (red) and of the partition {Ω i } (gray) in the case M (green) is being a H 1 -rectifiable set. Here µ + = δ x 0 and µ -= 1/3(δ x 1 + δ x 2 + δ x 3 ).

Figure 1 . 2 :

 12 Figure 1.2: Example of the neighborhoods of the segment [0, l] × {0}. On the left the case in which r ε = ε, on the right the case in which r ε = a ε > ε. The cyan region is R ε andI aε = R ε ∪ (Q rε (0, 0) ∪ Q rε (l, 0)). Remark that supp(ρ ε ) = B(0, ε).

Figure 1 . 3 :

 13 Figure 1.3: On the left: Graph of ϕ N iter obtained via the Alternate Minimization Algorithm 1. On the right: in red, one of the solutions to the Steiner problem, while in blue, a minimizer of the energy E β .

  Figure 1.4 and Figure 1.3 the joint minimization procedure allows to displace the functions. Indeed, as shown in Figure 1.5, the energy decreases during the joint minimization procedure as it. Let us propose a second Chapter 1. Affine cost function Algorithm 2 Joint Minimization Require:

Figure 1

 1 Figure 1.4: On the left the graph of ϕ N iter on the right the one of σ N iter obtained via the Joint Minimization Algorithm 2.

Figure 1 . 5 :

 15 Figure 1.5: Behaviour of the energy during the joint minimization iterations of Algorithm 2.

Figure 1 . 6 :

 16 Figure 1.6: Graph of the couple (σ N iter , ϕ N iter ) obtained via Algorithm 1 in the case of 3, 4, 5 and 6 points located on the vertices of a regular polygon.

Figure 1 . 7 :

 17 Figure 1.7: Graph of the exact solutions to the Steiner Problem constrained as in the previous figure.

Figure 2

 2 Figure 2.1: Illustration of the sections of B r , B r/4 and C r * . In grayscale we represent the level sets of the function χ r * (x )1 [-r,r] .

  s) has a meaning on the hyperplane {x 1 = s} in the sense of trace, moreover, g ε is continuous. Now, let us fix r ∈ [(1ξ)r * , r * ] such that µ({-r, r} × B r ) = 0 (which holds true for a.e. r ∈ [(1ξ)r * , r * ]) and let us define the mean value, ḡε := 1 2r r -r g ε (s) ds. Chapter 2. Multidimensional case From σ ε * σ, |σ ε | * µ, we have lim ε↓0 ḡε = 1 2r (-r,r)×B r * χ r * (x ) dσ(s, x ) • e 1 =: m.

Figure 2 . 2 :

 22 Figure 2.2: Illustration of the interval I and both its r and (r * ε)-enlargement for r * ≥ 1. In grayscale we plot the levels of the function ζ ε , whilst the striped region corresponds to the cylinder C r,ε .

  r and |x | ≤ r . Let χ(x ) be the radial cutoff introduced in the previous proposition and set χ r(x ) = χ(x /r), σ 0 ε = σ ε , e 1 ∧ • • • ∧ e k and for any x ∈ B r set g ε (x ) := B r χ r(x ) d σ ε , e I 0 = B r χ r(x ) dσ 0 ε , as in equation (3.8). Up to a smaller choice for r 0 we can assume B r ∩ S = ∅ therefore ∂σ B r = 0, and from equations (3.8) -(3.11) it holds

Proposition 4. 2 (

 2 Lower bound on reduced phase field functional). Let I = (a, b) ⊂ R and 0 ≤ δ ≤ η ≤ 1. Let I η ⊂ {x ∈ I | ϕ 1 (x), . . . , ϕ N (x) ≥ η}, and denote the collection of connected components of I \ I η by C η . Furthermore define the subcollection of connected components C ≥ η = {C ∈ C η | inf C ϕ 1 , . . . , inf C ϕ N ≥ δ} and C ≥ = C∈C ≥ η C. Finally assume ϕ i (a), ϕ i (b) ≥ η for all i = 1, . . . , N .

  x) else, where t C is chosen such that C | θ| dx = C |ϑ| dx (this is possible, since for t C = ∞ we have | θ| ≥ |ϑ| and for t C = -∞ we have | θ| ≤ |ϑ| everywhere on C). The cost did Chapter 4. Piecewise affine cost functions not change by this modification since

  Corollary 4.1 (Γlim inf inequality for reduced functionals). Let J ⊂ R be open and bounded, ϑ ∈ M(J), and ϕ 1 , . . . , ϕ N ∈ L 1 (Ω). Then Γlim inf ε→0 G ε /ϑ, ϕ 1 , . . . , ϕ N ; J) ≥ G (ϑ, ϕ 1 , . . . , ϕ N ; J)

  A, B, C ⊂ I (due to the subadditivity of h) and in the third inequality we used h(m) ≤ α 0 m as well as the lower semi-continuity of the mass on an open set. Letting now ζ → 0 (so that by the σ-continuity of ϑ we have |ϑ|(I \B ζ ({x 1 , . . . , x K })) → |ϑ|(I \ {x 1 , . . . , x K })) we obtain lim inf ε→0

2

  which implies |ϑ|(I \ {x 1 , . . . , x K }) = 0 and thus |ϑ ⊥ |(I) = 0 as well as S ϑ ∩ I ⊂ {x 1 , . . . , x K }. Next note that for all i ∈ {1, . . . , K} with |m ϑ (x i )| > 0 we have

Figure 4

 4 Figure 4.1: Left: Sketch of the optimal profile of |σ ε | and a phase field ϕ ε ῑ for some ῑ > 0 with m = 2, ε = 0.1, α ῑ = 1, β ῑ = 1. Right: Sketch of the numerical solution to the 1D problem with the same parameters.

  ε 2 , . . . , ϕ ε N = ϕ ε N . Reintroducing now the indices k and j,

Figure 4 . 4 :

 44 Figure 4.4: Computed mass flux σ and phase field ϕ for same parameters as in Figure 4.2.

  Figure 4.7.

Figure 4 . 7 :

 47 Figure 4.7: Computed mass flux σ and phase field ϕ 1 for a central point source and a spatially uniform sink outside a circle of radius and for the cost function shown on the right, ε = 0.005.

Figure 5 . 1 :

 51 Figure 5.1: Graphs of the function h in red and the corresponding h * in blue for the choices: h(x) = 2 √ x on the left and h(x) = 3x 1/3 on the right.

Lemma A. 2 (

 2 Approximation of u).There exists a sequence (w j ) ⊂ P C( Ω) with the following properties: a) w j → u weakly in BV ( Ω), b) supp w j ⊂ Ω, c) lim sup j→∞ E β (w j , 1) ≤ E β (u, 1), d) J w j is contained in a finite union of segments for any j ∈ N, e) |Dw j -Du|(∪∂Ω i ) → 0.

a

  ) there exists a constant C = C(m, β) ≤ C 0 (1 + √ βm) (m, r, r) < C and h d ε,β (m, r) < C. (B.2) b) Both the problem defined in equation (2.8) and equation (B.1) admit a minimizer.Moreover among the minimizers of G ε,β in Y ε,β (m, r) it is possible to choose a radially symmetric pair (ϑ ε , ϕ ε ) such that ϕ ε is radially non-decreasing and ϑ ε is radially non-increasing.

≥ Br\Br ξ ϕ ε |ϑ ε | 2 dx ε ≥ ξ Br\Br ξ |ϑ ε | dx 2 ω

 2 √ βm) 1/d ε ≤ r and (1 + ( √ βm) 1/d )ε ≤ r, we can set r 1 := ( √ βm) 1/d . We obtain, max{h d ε,β (m, r, r), h d ε,β (m, r)} ≤ C(1 + βm). B.1. Auxiliary problem b) To show the existence of minimizers for both minimization problems we use the direct method of the Calculus of Variation. The lower semicontinuity of the integral with integrand u|ϑ| 2 is ensured by Ioffe's theorem [AFP00, theorem 5.8]. Now given any minimizing pair ( θε , φε ) ∈ Y ε,β (m, r), let ϑ ε be the decreasing Steiner rearrangement of θε and ϕ ε the increasing rearrangement of φε . Indeed, since φε has range in [η, 1], we still have ϕ ε |∂Br ≡ 1. Polya's Szego and Hardy-Littlewood's inequalities [Tal76, LL97] ensureG ε,β (ϑ ε , ϕ ε ) ≤ G ε,β ( θε , φε )Let us prove the asymptotic equivalence of the values h d ε,β (m, r, r) andh d ε,β (m, r) as ε ↓ 0.Lemma B.2 (Equivalence of the two problems). For any r < r and m > 0 it holds|h d ε,β (m, r, r)h d ε,β (m, r)| ε↓0 -→ 0 Proof. Step 1: [h d ε,β (m, r, r) ≤ h d ε,β (m, r) + O(1)] Consider for each ε the radially symmetric and monotone pair (ϑ ε , ϕ ε ) ∈ Y ε,β (m, r) as introduced in the previous lemma. Take ξ ∈ (η, 1) and let us set r ξ := sup{t ∈ (0, r) : ϕ ε (t) ≤ ξ} with r ξ = 0 if the set is empty. (B.3) By Cauchy-Schwartz inequality it holds C d r d ε . Let us define ∆ ξ := Br\Br ξ |ϑ ε |, the latter ensures that ∆ ξ ∈ o(ε 1/2 ). Let us now set θε = mϑε Br ξ ϑε

  (m, r) ≥ h d β (m) then equation (2.11) easily follows from Lemma B.2. For m = 0 set ϑ = 0 and u = 1, then (ϑ, ϕ) ∈ Y ε,β (0, r) for any radius r and G ε,β (ϑ, ϕ; B r ) = 0 for each ε. Now suppose m > 0 and let ξ ∈ (η, 1). Consider the radially symmetric and monotone minimizing pair (ϑ ε , ϕ ε ) of Lemma B.1 and r ξ introduced in equation (B.3). Let us split the set of integration in the two sets B r ξ and B r \ B r ξ , we obtain

m ω d

 d (εr * ) d on the ball B εr * and zero outside. Indeed, the pair (ϑ ε , ϕ ε (|x|)) belongs to Y ε,β (m, r). That is because ϕ ε is greater then η and attains value 1 at the border of B r andBr ϑ ε (x) dx = m ω d (εr * ) d ω d (εr * ) d = m.

  Figure D.2: Steiner Minimal Tree reliant 10000 points répartis aléatoirement dans le plan. Le problème a été résolu en utilisant l'algorithme GeoSteiner [WZ97], qui est actuellement l'algorithme exact le plus efficace pour calculer les arbres Steiner minimums.

Figure D. 4 :

 4 Figure D.4: Exemple d'une surface enjambant une frontière 1-dimensionnelle composée de trois cercles orientés.

  D.4) Par White dans [Whi99a, 6] les conditions (D.3) sont suffisantes pour étendre E h sur M(Ω, R n ). En choisissant h(m) = |m| dans l'équation (D.4) on obtient le fonctionnelle de masse qui associe à chaque vecteur σ sa variation totale |σ| = sup{(ϕ, σ) : ϕ ∈ C 0 (Ω, R n ), ϕ ∞ ≤ 1}. Sinon, avec h(m) = χ {m =0} où χ désigne la fonction caractéristique d'un ensemble, E h se réduit au fonctionnelle de taille qui mesure la longueur du support de σ, à savoir σ → H 1 (supp(σ)). D'autres choix remarquables sont représentés dans la Figure D.5.

  dans l'équation (D.7), un k-courant σ est dit k-rectifiable si nous pouvons lui associer un triplet (θ, τ, Σ) de telle sorte que σ, ω = Σ θ ω, τ dH k où Σ est un sous-ensemble k-rectifiable de Ω, τ à H k a.e. point est un simple k-vecteur unitaire qui enjambe le plan tangent à Σ et θ est une fonction L 1 (Ω, H k Σ) qui peut être supposée positive. L'espace vectoriel de Courants rectifiables est indiqué par R k (Ω). Parmi ceux-ci nous nommons le sous-ensemble P k (Ω) de courants rectifiables pour lesquels Σ est une union finie de polyèdres et θ est constant sur chacun d'eux, ceux-ci seront appelés Chaînes polyédriques. Pour tout k-courant σ tel que σ et ∂σ sont de masse finie, nous disons que σ est un k-courant normal et nous écrivons σ ∈ N k (Ω). Sur l'espace D k (Ω) nous pouvons définir la norme flat parF(σ) = inf {|σ R | + |σ S | : σ = σ R + ∂σ S où σ S ∈ D k+1 (Ω) et σ R ∈ D k (Ω)} ,qui métrifie la topologie faible- * pour les courants de N k (Ω) avec support compact. Enfin, l'espace de flat chains F k (Ω) consiste en la fermeture de P k (Ω) dans la topologie F. Par le schéma de Federer [Fed69, 4.1.24] nous avons la suivante chaîne d'inclusionsP k (Ω) ⊂ N k (Ω) ⊂ F k (Ω).Suivant la stratégie proposée par Fleming[START_REF] Federer | Normal and integral currents[END_REF][START_REF] Fleming | Flat chains over a finite coefficient group[END_REF] dans le contexte des flat chains à coefficients en groupes, nous définissons maintenant l'énergie E h sur l'espace des flat chains. Soit h est une fonction de coût de transport et σ = (m i τ i , Σ i ) un courant polyédrique on imposeE h (σ) := i h(m i )H k (Σ i ).Par analogie à ce qui a été fait avant, nous étendions E h sur l'espace des chaînes plates par relaxationE h (σ) := inf lim inf j→∞ E h (σ j ) : σ j polyhedral et F(σ jσ) → 0 .Dans le chapitre 3 nous cherchons des approximations aux problèmes de type min{Eh (σ) : ∂σ = ∂σ 0 } (D.10) où σ 0 est unek-courant polyédrique donnée. Ces problèmes ont été introduits et étudiés dans [Mor89, DPH03] par Morgan, De Pauw et Hardt entre autres pour proposer différents modèles pour des surfaces minimales de film de savon. Ce dernier est la généralisation k-dimensionnelle du problème de minimisation défini dans (D.6). Comme indiqué dans [Whi99a, Whi99b] E h a une formulation explicite pour les courants rectifiables, à savoir pour un courant rectifiable (m, τ, Σ), nous avons E h (σ) := Σ h(m) dH k . Ce résultat a été prouvé en [CDRMS17, Proposition 2.6], et est la conséquence du théorème d'approximation polyédrique suivant (UP) Inégalité Γlim sup: pour tout x ∈ X il existe une séquence (x ε ) ⊂ X telle que xε → x et lim sup ε→0 F ε (x ε ) ≤ F (x).

  Theorem D.6 (Caractérisation pour la Γ-convergence). Soit X un espace métrique,et ε > 0, soit F ε : X → [0, +∞] et F : X → [0, +∞]. F ε Γ -→ F si et seulement si pour chaque G fonctionnel continu, si x ε minimise F ε + G et x ε → x alors x minimise F + G .

  d ϕ (x, y) = inf γ ϕ dH 1 : γ ∈ C([0, 1], Ω), γ(0) = x, γ(1) = y .La distance d ϕ (x, y) est nulle si et seulement si les deux points x, y sont reliés par un chemin sur lequel ϕ est égal à 0. Considérez la fonctionelle Ω x 0 , x i ) oú c ε → 0 quand ε → 0. Remarquez d'abord que si N i=1 d ϕ (x 0 , x i ) = 0 (D.11) alors l'ensemble {ϕ = 0} devrait inclure un sous-ensemble connecté pour chemins contenant {x 0 , .. . , x N }. L'argument heuristique pour le résultat de Γ -convergence suit les idées présentées dans le cas de la fonctionnelle de Ambrosio-Tortorelli. Le résultat exact dans [BLS15] est La preuve du résultat de convergence Γ est obtenue pour Ω des sous-ensembles convexes et ouverts de R 2 . L'avantage de choisir une pénalisation quadratique en σ est que le problème de Lagrange augmenté associé au fonctionnel peut àtre explicitement résolu dans la variable dual. Il est donc possible de concevoir un algorithme de minimisation alterné composé de deux fonctionnelles elliptiques et lisses résolubles par des méthodes d'éléments finis. The algorithm is proposed and studied at the end of the chapter. L'algorithme est proposé et étudié à la fin du chapitre. De plus nous présentons et étudions d'autres algorithmes qui profitent d'un concept de 'dérivée de forme' pour améliorer la qualité de l'approximation. La généralisation à Ω ⊂ R n est traitée dans le Deuxième Chapitre. Pour obtenir le résultat dans une dimension supérieure, la composante Modica-Mortola de la fonctionnelle doit àtre recalibrée. Comme observé dans [Ghi14] cela conduit à l'introduction de certaines non linéarités dans le fonctionnelle comme suitΩ ϕ|σ| 2 ε dx + Ω ε p-n+1 |∇ϕ| p + (1ϕ) 2 ε n-1 dx, (D.15)pour quelques p > n -1. Encore une fois σ est corrélé avec la contrainte de divergence (D.13) pour un choix approprié de ρ ε et il faut imposer une borne inférieure pour les fonctions de champ de phase, à savoir ϕ ≥ βε n . Nous prouvons la Γ-convergence de la famille des fonctionnelles ci-dessus vers E h n-1 β oú la fonction de coàt h n-1 β est la limite en ε d'un problème d'optimisation dépendant de la codimension n -1. Plus précisément pour une balle B r ⊂ R n-1 nous définissonsh ϕ) ∈ W 1,p (B r ), ϕ = 1 sur ∂B r et Br θ dx = m.Ce dernier problème d'optimisation correspond à la version 0-dimensionnelle de (D.15). Nous introduisons et étudions h d ε,β (obtenu en remplaçant n-1 par d dans cette dernière formule) dans l'annexe. Bouchitté, Dubs et Seppecher in[START_REF] Bouchitté | Transitions de phases avec un potentiel dégénéré à l'infini, application à l'équilibre de petites gouttes[END_REF] ont étudié des problèmes similaires de transition de phase avec contrainte de masse qui conduisent à des mesures concentrées sur des atomes dans le contexte de l'équilibre des gouttelettes. En particulier, nous montrons que h d β est indépendant de r et qu'il s'agit d'une fonction de coàt de transport satisfaisant aux conditions (D.3). Nous prouvons également qu'il existe une constante c > 0 m > 0.Remarquez que la composante Modica-Mortola de la composante fonctionnelle étudiée dans le deuxième chapitre dépend de n -1, la co-dimension du problème dans le cas de mesures rectifiables. Dans le Chapitre Trois, nous étudions un rééchelonnement différent pour approcher les minima de (D.10) définis pour k-courants, à savoirΩ ϕ|σ| 2 ε dx + Ω ε p-n+k |∇ϕ| p + (1ϕ) 2 ε n-k dx. (D.16)Dans ce contexte, σ n'est plus une mesure vectorielle, pour tenir compte de la frontière, la contrainte doit àtre modifiée de façon appropriée. Soit σ 0 une k-courant polyédrique donnée, pour ρ ε une suite régularisante standard, nous laissons σ àtre un k-courant telle que ∂σ = ∂σ 0 * ρ ε .(Dans l'équation (D.16), le courant est identifié avec sa densité.) Dans le chapitre nous introduisons formellement l'énergie et montrons qu'elle Γ-converge à l'énergie E h définie dans (D.10) pour la fonction de coàt de transport h = h n-k β étudié dans l'appendice.

FFigure D. 6 :

 6 Figure D.6: Computed mass flux σ and phase fields ϕ 1 , ϕ 2 , ϕ 3 for the cost function shown on the right, ε = 0.005. The color in σ indicates which phase field is active. The result is obtained by optimizing the functional defined in Chapter Three.

Titre:

  Approximations par champs de phases pour des problemes de transport branch éMots cl és : transport branch é, calcul des variations, th éorie g éom étrique de la mesure, Γ-convergence. R ésum é : Dans cette th èse, nous concevons des approximations par champ de phase de certains probl èmes de Transport Branch é. Le Transport Branch é est un cadre math ématique pour mod éliser des r éseaux de distribution offre-demande qui pr ésentent une structure d'arbre. En particulier, le r éseau, les usines d'approvisionnement et le lieu de la demande sont mod élis és en tant que mesures et le probl éme est pr ésent é comme un probl éme d'optimisation sous contrainte. Le co ût de transport d'une masse m le long d'un bord de longueur est h(m) et le co ût total d'un r éseau est d éfini comme la somme de la contribution sur tous ses arcs. Le cas du Transport Branch é correspond avec la choix h(m) = α|m| o ù α est dans [0, 1). La sous-additivit é de la fonction cout s'assure que d éplacer deux masses conjointement est moins cher que de le faire s épar ément. Dans ce travail, nous introduisons diverses approximations variationnelles du probl ème du transport branch é. Les fonctionnelles que on vais utiliser sont bas ées sur une repr ésentation par champ de phase du r éseau et sont plus lisses que le probl ème original, ce qui permet des m éthodes d'optimisation num érique efficaces. Nous introduisons une famille des fonctionnelles inspir ées par le fonctionnelle de Ambrosio et Tortorelli pour mod éliser une fonction de co ût h affine dans l'espace R 2 . Pour ce cas, nous produisons un r ésultat complet de Γ-convergence et nous le corr élons avec une proc édure de minimisation altern ée pour obtenir des approximations num ériques des minimiseurs. Puis nous g én éralisons cette approche à n'importe quel espace R n et obtenons un r ésultat complet de Γ-convergence dans le cas de surfaces k-dimensionnelles avec k < n. En particulier, nous obtenons une approximation variationnelle du probl ème du Plateau dans n'importe quelle dimension et co-dimension. Dans la derni ère partie de la th èse, nous proposons deux approches g én érales pour des fonctions de co ût concave. Dans le premier, nous introduisons une approche par plusieurs champs de phase et r écup érons n'importe quelle fonction de co ût affine par morceaux. Enfin, nous proposons et étudions une famille de fonctions permettant d'obtenir dans la limite toutes fonction de co ût concave h. Title : Phase-field approximations for some branched transportation problems Keywords : branched transport, calculus of variations, geometric measure theory, Γ-convergence.

  therefore for any O ∈ A S ( Ω) applying the localization described in Section 1.3 we can associate to each σ ε a function u ε ∈ W 1,2 (O) with mean value 0 such that σ ε = ∇u ⊥ ε in O.

	Since
	|∇u ⊥ ε | = |∇u ε | by Theorem 1.5 there exists a u ∈ P C(O) such that, up to extracting a subsequence, u ε * u. Eventually, from formula (1.8), we get

  Consider the countable family of sets {O i } ⊂ A S ( Ω) made of the relatively open rectangles O i ⊂ Ω \ S with vertices in Q 2 . The local result stated in Theorem 1.5 gives for any i ∈ N

  I r * ε and they are oriented by the vector e 1 therefore |σ 1 ε

both σ 1 ε and σ 2 ε are obtained 2.5. Γ-limsup inequality trough convolution it holds supp(σ 1 ε ) ∪ supp(σ 2 ε ) ⊂

  ϕ 1 , . . . , ϕ N ; I), as desired. Let us now abbreviate m 0 = (Iη∪C ≥ )\K ε 0 |ϑ| dx. Using the definition of ω ε as well as

  Graph of h and two of its approximations h k 1 and hk 2 with k 1 < k 2 .

				h(t),				for t ≤ 2 -k , otherwise.
	2
	1.8
	1.6
	1.4
	1.2
	1
	0.8
	0.6
	0.4
										f
	0.2									f k1
										f k2
	0
	0	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09	0.1
	Figure A.1:

  5 Proof of Proposition 2.3

	Propositions 2.1, 2.2 and lemma B.2 ensure that
	h d β (m) = lim

  Présentons précisément le schéma du Transport Branché[START_REF] Bernot | Optimal transportation networks[END_REF][START_REF] Villani | Topics in optimal transportation[END_REF]. Tout d'abord, nous introduisons les réseaux de transport dans un ensemble ouvert Ω ∈ R n , et le fonctionnelle de coût associé. Pour cela, considérez un segment Σ ⊂ Ω, un nombre réel positif m ∈ R + et un vecteur τ ∈ S n-1 tangent à Σ, l'écriturem τ H 1 Σ (D.2)définit une mesure à valeur vectorielle, où H 1 Σ est la mesure de Hausdorff 1-dimensionnelle en R n limitée au segment Σ. Intuitivement, la mesure Radon H 1 Σ associe à tout ensemble mesurable A la longueur de A ∩ Σ. Nous disons qu'une mesure vectorielle σ ∈ M(Ω, R n ) est polyhedral si c'est une somme finie de mesures de la forme (D.2), à savoir σ

  , Théorème 2.7] le même auteur analyse le problème de la régularité. Pour énoncer le résultat, nous devons introduire la notion de rectifiable vector measure. A savoir une mesure vectorielle σ est dit rectifiable si σ = m τ H 1 Σ (D.7) où Σ, le support de σ comme distribution, est un ensemble H 1 -rectifiable, sa densité H 1 est la fonction m ∈ L 1 (H 1 Σ) et τ : Σ → S n-1 génère pour H 1 -a.e. point dans Σ l'espace tangent à Σ. Dans ce qui suit, nous dénotons avec (m, τ, Σ) la mesure rectifiable σ définie dans (D.7). (Structure des réseaux d'énergie finie). Pour 0 ≤ α < 1 si σ ∈ X µ + ,µ - est de variation totale finie et d'énergie E h finie alors il est rectifiable. De plus si σ = (m, τ, Σ) nous avons E représentation explicite de la fonctionnelle à toute mesure rectifiable. Le cas des fonctions génériques de coût de transport a été pris en considération par Brancolini et Wirth in [BW18, Proposition 2.32] qui montre que Proposition D.1 ( Énergie Gilbert-Steiner généralisée ). Soit µ + , µ -

	Theorem D.2

h (σ) = Σ |m| α dH 1 . (D.8) L'équation (D.8) est particulièrement significative puisqu'elle étend la

Ω(ug) 2 dx.

Remerciements

Chapter 4. Piecewise affine cost functions Algorithm 6 Minimization for N > 1, α 0 < ∞ function MPFSD(ε start , ε end , N iter , α 1 , . . . , α N , β 1 , . . . , β N , µ + , µ -, ρ ε end ) set f ε = (µ +µ -) * ρ ε end set (σ 0 , •, λ 0 ) = SP F S(ε start , ε end , N iter , α 1 , β 1 , µ + , µ -, ρ ε end ) compute regions R ε 0 , Rε 1 , . . . , Rε N via (4.12) for j = 1, . . . , N iter do set ε j = ε start -(j -1) εstart-ε end N iter -1 set ϕ j i to the minimizer of (4.13) for given fixed σ = σ j-1 , i = 1, . . . , N update regions R ε 0 , Rε 1 , . . . , Rε N via (4.8) and (4.12) set γ j ε = min i=1,...,N (ϕ j i ) 2 + α 2 i ε 2 /β i set (σ j , λj ) = (σ j-1 , λj-1 ) -DR(σ j-1 , λj-1 ) -1 R(σ j-1 , λj-1 ) for γ ε = γ j-1 ε end for end function return σ N iter , ϕ N iter Appendix A

Density result for vector measures in R 2

We show that measures which have support contained in a finite union of segments, are dense in energy. Without loss of generality let us assume that σ ∈ M S (Ω) is such that

From the above properties, we can write

for some u ∈ P C(Ω). Our strategy is the following, using existing results [START_REF] Bellettini | The Γ-limit for singularly perturbed functionals of Perona-Malik type in arbitrary dimension[END_REF], we build an approximating sequence for u on each Ω j whose gradient is supported on a finite union of segments. We then glue these approximations together to obtain a sequence (w j ) approximating u in Ω. Where Ω is an open set containing Ω. The main difficulty is to establish that Dw

for h a continuous, sub-additive and increasing function on [0, +∞) such that h(0) = 0 and lim t→0 h(t) t = +∞. Then there exists a sequence (u l ) ⊂ P C(Ω) with the following properties:

• J u l is contained in a finite union of facets of polytopes for any h ∈ N. In particular for any n ∈ N,

For each fixed k ≥ 0, let us introduce the function

where t denotes the integer part of the real t. Note that

Indeed, u k → u strongly in BV ( Ω), as H 1 (J u ) < +∞. Moreover, we see that

Step 2: Let us approximate the function u k . Let us fix k ≥ 0 and Ω i . We can apply Lemma A.1 to the function u k Ω i and to the energy E h k (•, Ω i ). We obtain a sequence (w i j ) which enjoys the following properties:

is contained in a finite union of segments for any j ∈ N,

Let us now define globally w j :=

From the above properties, we have

Eventually, using a diagonal argument, we have proved the existence of a sequence (w j ) ⊂ P C( Ω) satisfying claims (a), (b) and (d) of the lemma. Moreover, item (c) is the consequence of (A.2) and (A.3) and item (e) follows from (A.1) and (A.4).

Going back to the H 1 -rectifiable measures σ = U (m σ , τ σ , Σ σ ), we define the sequence

We deduce from the previous lemma: Lemma A.3. There exists a sequence (σ j ) ∈ M S (Ω) with the properties:

-σ j → σ with respect to weak- * convergence of measures, -σ j = U (m σ j , τ σ j , Σ σ j ) with M σ j contained in a finite union of segments, -lim sup j→∞ E β (σ j , 1) ≤ E β (σ, 1).

Appendix B

Reduced problem in dimension nk

B.1 Auxiliary problem

In this appendix we show the results previously stated in Section 2.2 of Chapter 2, with the notation introduced therein let us define the auxiliary set

and the associated minimization problem

For the sake of clarity let us recall that the functional introduced in equation (2.7) has the expression

Analogous optimization problem to (B.1) with mass constraint appears in models of droplets equilibrium. Bouchitté et al. in [START_REF] Bouchitté | Transitions de phases avec un potentiel dégénéré à l'infini, application à l'équilibre de petites gouttes[END_REF] study a one dimensional smooth version of the problem in which the mass constraint is on the phase field variable ϕ. Minimizing (B.1) in ϑ we obtain a functional depending only on the variable ϕ which can be interpreted as a variant in higher dimension of the cited work. The outline of the appendix is the following. First we show that both h d ε,β (m, r, r) and h d ε,β (m, r) are bounded by the same constant as ε ↓ 0 and that the value of the second term is achieved by a radially symmetric pair of Y ε,β (m, r). These two facts are then used to show that for each m the limit values of h

) is a minimum. Moreover there is a unique function achieving the minimum which is nondecreasing with range in the interval [ξ, 1]. Finally the function q d satisfies the following properties

Recalling the definition (2.10) of q d ∞ , we have q d ∞ (ξ, r) = q d (ξ, r 1 , ∞), and q d ∞ (0, 0) > 0. Furthermore for any r > 0 the map ξ → q d ∞ (ξ, r) is convex and continuous on (0, +∞).

Proof. Let (ξ, r 1 , r 2 ) ∈ [0, 1]×R + ×R + , the infimum is actually a minimum by means of the direct method of the calculus of variations. Such minimum is absolutely continuous on the interval (r 1 , r 2 ) by Morrey's inequality and is unique since

As a consequence for every minimizer of (B.7) we have ξ ≤ v ≤ 1. Similarly setting now study the monotonicity of q d . To do so let v be the minimizer for (ξ, r 1 , r 2 ):

1. Let r 2 > r 2 and let us extend v by 1 on the interval (r 2 , r 2 ). We have

Hence r 2 → q d is nonincreasing.

Let v be the minimizer of (B.7) and v(s

Therefore r 1 → q d is nondecreasing.

3. Let 0 ≤ ξ < ξ ≤ 1 and v the absolutely continuous, nondecreasing minimizer of problem q d (ξ, r 1 , r 2 ). Then there exists r ∈ (r 1 , r 2 ) for which v(r) = ξ. Hence

Hence, ξ → q d is nonincreasing. Finally, for ξ = 1 consider the constant function v ≡ 1 to get q d (1, r 1 , r 2 ) = 0.

Indeed, in view of the monotonicity, for every r 1 and r 2 we have

Let us show q d ∞ (0, 0) > 0. As a matter of facts, taken the minimizer v for the problem (2.10), there exists r ∈ (0, +∞) such that v(r) = 1/2 and we have

A direct evaluation gives min

and we obtain the estimate

Lastly, let us show that for any r the function q d ∞ (•, r) is convex. Consider two values ξ 1 , ξ 2 ∈ (0, 1) and the associated minimizers v 1 , v 2 for the respective energy q d ∞ (•, r).

Let us now prove the sub-additivity. For a radius r consider the competitors (ϑ j , u j ) ∈ Y ε,β (m j , r) for j = 1, 2. Consider the ball B 2r+1 centered in the origin and two points x 1 , x 2 such that the balls B r (x 1 ), B r (x 2 ) are disjoint and contained in B 2r+1 . Set

and observe that the pair (ϑ, u) belongs to Y (m 1 + m 2 , 2r + 1). Being the balls B r (x j ) disjoint we have

Passing to the limit as ε ↓ 0, and recalling that it is independent of the choice of the radius, we get

We conclude the appendix by showing that Lemma B.4. For any sequence β i ↓ 0 it holds

pointwise.

Proof. We have already shown that

Finally simply recall that (d -1) ω d q d ∞ (0, 0) = κ and that q d ∞ (0, •) is continuous.

Appendix C

Slicing of measures

We derive now some technical construction for divergence measure vector fields which are needed to reduce the Γlim inf inequality of Chapters 4 and 5 to the lowerdimensional setting. In particular, we will introduce slices of a divergence measure vector field, which in the language of geometric measure theory correspond to slices of currents. We will slice in the direction of a unitary vector ξ ∈ S n-1 with orthogonal hyperplanes of the form

The orthogonal projection onto H ξ,t is denoted

The slicing will essentially be performed via disintegration. Let σ be a compactly supported divergence measure vector field. By the Disintegration Theorem [AFP00, Thm. 2.28], for all ξ ∈ S n-1 and almost all t ∈ R there exists a unique measure ν ξ,t ∈ M(H ξ,t ) such that

We decompose π ξ # |σ • ξ| into its absolutely continuous and singular part according to

for dt the Lebesgue measure on R.

Lemma C.1. For any ξ ∈ S n-1 and any compactly supported divergence measure vector field σ we have σ ⊥ ξ = 0, that is, the measure π ξ # |σ • ξ| = σ ξ (t) dt is absolutely continuous with respect to the Lebesgue measure on R. Moreover, for almost all t ∈ R and any compactly supported θ ∈ C ∞ (R n ) we have

Proof. Abbreviate H = ξ ⊥ = H ξ,0 with corresponding orthogonal projection π H , let φ ∈ C ∞ (H) and ψ ∈ C ∞ (R) be compactly supported, and define

Introducing Ψ(t) = t -∞ ψ(s) ds we obtain via the chain and product rule

so that (denoting by χ A the characteristic function of a set A)

(Note that we could just as well have used χ {ξ•x>s} instead of χ {ξ•x≥s} , which would ultimately lead to integration domains {ξ • x ≤ t} in (C.1); for almost all t this will be the same.) Applying the Fubini-Tonelli Theorem we obtain

where in the second step we just added 0

in the square brackets. On the other hand, using the disintegration of σ • ξ we also have

Comparing both expressions for I(φ, ψ) we can identify

Since the right-hand side has no singular component with respect to the Lebesgue measure, we deduce H ξ,s φ(π H (y)) dν ξ,s (y) σ ⊥ ξ (s) = 0. Now note that any compactly

Remark 10 (Characterization of divergence measure vector fields). By a result due to Smirnov [START_REF] Smirnov | Decomposition of solenoidal vector charges into elementary solenoids, and the structure of normal one-dimensional flows[END_REF], any divergence measure vector field σ can be decomposed into simple oriented curves σ γ = γ # γ ds [0, 1] with γ : [0, 1] → R n a Lipschitz curve and ds the Lebesgue measure, that is,

with J the set of Lipschitz curves and µ σ a nonnegative Borel measure. The results of this section can alternatively be derived by resorting to this characterization, since the slice of a simple oriented curve σ γ can be explicitly calculated.

Appendix D Résumé substantiel en langue française

Lors de la conception d'un réseau de distribution offre-demande, il convient de lui donner une structure d'arbre dans laquelle il est préférable de regrouper la masse dans le processus de transport. Cette hypothèse émerge de nombreuses observations, par exemple, la structure des vaisseaux sanguins dans le système cardiovasculaire est requise pour distribuer le sang d'une source concentrée dans le coeur à un volume répandu ou vice-versa, le système racinaire d'un arbre a besoin de récupérer l'eau du sol. Dans ces situations, nous pouvons observer à quel point des vaisseaux larges et longs sont préférables plutôt que des vaisseaux éparpillés. L'hypothèse que nous faisons est que le réseau observé est optimal par rapport à un coût donné parmi tous les réseaux possibles se développant à partir d'une source et irriguant un puits donné. Ces structures apparaissent dans un large gamme de situations et de nombreux efforts ont été faits par la communauté mathématique afin de donner un modèle précis capable de décrire toutes les caractéristiques observables de ces réseaux.

En Nous allons profiter de ce théorème dans le contexte des mesures vectorielles, avec la notation introduite ci-dessus le même théorème se lit comme suit Theorem D.5 (Rectifiability for vector valued measures). Soit σ ∈ M(Ω, R n ). Si |σ|(Ω) + |∇ • σ|(Ω) < ∞, ∇ • σ est une somme dénombrable de masses Dirac et il existe un ensemble de Borel Σ avec H 1 (Σ) < ∞ et σ = σσ Σ alors σ est une mesure vectorielle rectifiable.

Approximation variationnelle pour les problèmes de minimisation

Nous fournissons des approximations aux problèmes définis dans (D.6) dans le sens de Γ-convergence. Cette dernière est une notion de convergence fonctionnelle introduite par De Giorgi [START_REF] Giorgi | Sulla convergenza di alcune successioni d'integrali del tipo dell'area[END_REF] pour traiter des problèmes variationnels. En suivant [DM93, Bra98, AD00, Bra02] on donne la définition opérationnelle de Γ-convergence.

Definition 4 (Γ-convergence). Soit X un espace métrique, et pour ε > 0 on donne F ε : X → [0, +∞]. Nous disons que F ε Γ converge vers F sur X en tant que ε → 0 et nous notons F ε Γ -→ F si les deux conditions suivantes sont satisfaites :

Dans ce qui précède BV(Ω) indique l'espace de ces fonctions ϕ tel que ϕ ∈ L Un autre exemple vient de l'approximation du fonctionnelle de Mumford-Shah pour la segmentation d'images. Dans [START_REF] Mumford | Optimal approximation by piecewise smooth functions and associated variational problems[END_REF] les auteurs considèrent une fonction g, définie sur un domaine Ω, représentant le niveau de gris d'une image d'un groupe d'objets donnée par une caméra, avec des discontinuités long les bords des objets. L'idée est que l'image segmentée u devrait être suffisamment lisse à l'extérieur d'un ensemble (n-1)-dimensionnel contenant l'ensemble de discontinuité K, à savoir u ∈ W 1,2 (Ω\K), et ce dernier devrait être choisi de H n-1 -size minimal. C'est pourquoi ils proposent d'optimiser dans les variables (u, K) l'énergie

The parameters α, β control the weight between the fidelity term |u-g| 2 and the size of the discontinuity set K. It is convenient to recast the problem in its weak formulation letting u ∈ BV(Ω) and replacing the set K with J u obtaining the functional S (u) := Ω |∇u| 2 + α(ug) 2 dx + βH n-1 (J u ).