Idriss Mazari
email: idriss.mazari@sorbonne-universite.fr

Quantitative inequality for the eigenvalue of a Schrödinger operator in the ball

Keywords: Stability for eigenvalues, Schrödinger operator, mathematical biology AMS classification: 35J15, 35Q93, 47A75, 49R05, 49Q10

The aim of this article is to prove a quantitative inequality for the first eigenvalue of a Schrödinger operator in the ball. More precisely, we optimize the first eigenvalue λ(V ) of the operator Lv := -∆ -V with Dirichlet boundary conditions with respect to the potential V , under L 1 and L ∞ constraints on V . The solution has been known to be the characteristic function of a centered ball, but this article aims at proving a sharp growth rate of the following form: if

for some C > 0. The proof relies on two notions of derivatives for shape optimization: parametric derivatives and shape derivatives. We use parametric derivatives to handle radial competitors, and shape derivatives to deal with normal deformation of the ball. A dichotomy is then established to extend the result to all other potentials. We develop a new method to handle radial distributions and a comparison principle to handle second order shape derivatives at the ball. Finally, we add some remarks regarding the coercivity norm of the second order shape derivative in this context.

In the first part (Section 1.2) of the introduction, we lay out the mathematical setting of this article, the main results and give the relevant definitions. In the second part of the Introduction, we give bibliographical references concerning quantitative inequalities and the biological motivations this study stems from. We then explain, in Subsection 2.1, how the proof differs from that of other quantitative inequalities. The core of this paper is devoted to the proof of Theorem 1. In the conclusion, we give a conjecture and a final comment on a possible way to obtain an optimal exponent using parametric derivatives.

Mathematical setting

The optimization of eigenvalues of elliptic operators defined on domains with a zero-order term (i.e with a potentiel) with respect to either the domain (with a fixed potential defined on a bigger domain) or the potential (with a fixed domain) is a classical question in optimization under partial differential equations constraints. The example under scrutiny here is the operator

L V : u ∈ W 2,2 0 (Ω) → -∆u -V u, (1)
where Ω is a smooth domain. Under the assumption that V ∈ L ∞ (Ω), this operator is known to have a first, simple eigenvalue, denoted by λ(V ), and associated with an eigenfunction u V , which solves

   -∆u V -V u V = λ(V )u V in Ω, u V = 0 on ∂Ω.
This problem has drawn a lot of attention from the mathematical community over the last decades, and is quite a general one. It is particularly relevant in the context of mathematical biology, see Section 1.4.2.

Main results

Notations

Here and throughout, the underlying domain is Ω = B(0; R).

The parameter r * is chosen so that

V * := χ B(0;r * ) = χ E * ∈ M(Ω).
The constant c n > 0 is the (n -1)-dimensional volume of the unit sphere in dimension n.

For any E ⊂ B, we define λ(E) := λ(χ E ).

Quantitative inequality

A classical application of Schwarz' rearrangement shows that V * is the unique minimizer of λ in M(Ω): ∀V ∈ M(Ω) , λ(V ) λ(V * ).

We refer to [START_REF] Kawohl | Rearrangements and Convexity of Level Sets in PDE[END_REF] for an introduction to the Schwarz rearrangement and to [START_REF] Lamboley | Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions[END_REF] for its use in this context. For the sake of completeness, we also prove this result in Annex A. The goalf of this paper is to establish the following quantitative spectral inequality:

Theorem 1. Let n = 2. There exists a constant C > 0 such that, for any V ∈ M(Ω), there holds

λ(V ) -λ(V * ) C||V -V * || 2 L 1 . (6)
Remark 1. The hypothesis n = 2 is essentially needed in Step 4 of the proof, where we gather estimates on second order shape derivatives and thus need to diagonalise the shape Hessian using Fourier series. This part of the proof can be adapted to dimension 3 using spherical harmonics, but we choose to present it in this low dimension context for the sake of readability. Furthermore, in

Step 5, we need convergence of the eigenfunctions in Hölder spaces and can obtain it in dimensions 2 and 3 in a straighforward manner.

We can rephrase this result in terms of Fraenkel asymmetry: as we will explain in Subsection 1.4.1, this is the natural property to expect in the context of quantitative inequalities. Indeed, if

V = χ E ∈ M(Ω), then ||V -V * || L 1 = |E * ∆E|
where ∆ stands for the symmetric difference, and, if we define the fraenkel asymmetry of E as

A(E) := inf B(x;r) ,χ B(x;r) ∈M(Ω)

|E∆B(x; r)| then it follows from Theorem 1 that

λ(χ E ) -λ(χ E * ) CA(E) 2 .
We thus have a parametric version of a quantitative inequality with what is in fact a sharp exponent.

A comment on parametric and shape derivatives

The proof of Theorem 1 relies on parametric and shape derivatives, and the aim of this Section is to give possible links between the two notions and, most notably, to give a situation where this link is no longer possible. This is obviously in sharp contrast with classical shape optimization, since here we are only optimizing with respect to the potential, while it is customary to derive Faber-Krahn type inequalities, i.e to optimize with respect to the domain Ω itself, see Subsection 1.4.1.

Roughly speaking, there are two ways to tackle spectral optimizatoin problems such as [START_REF] Belgacem | Elliptic Boundary Value Problems with Indefinite Weights, Variational Formulations of the Principal Eigenvalue, and Applications[END_REF]: the parametric approach and the shape derivative approach. By parametric approach we mean the following:

Definition 1. We define, for any V ∈ M(Ω), the tangent cone to M(Ω) at V as

T V := {h : Ω → [-1; 1] , ∀ε 1 , V + εh ∈ M(Ω)}
and define, provided it exists, the parametric derivative of λ at V in the direction h ∈ T V as

λ(V )[h] := lim t→0 λ(V + th) -λ(V ) t .
In this case, the optimality condition reads

∀h ∈ T V , λ(V )[h] 0.
In the conclusion of this article, we will explain why there holds

∀h ∈ T V * , λ(V * )[h] C||h|| 2 L 1 . Since, for any V ∈ M(Ω), h := V -V * ∈ T V * , what we get is λ(V * )[V -V * ] C||V -V * || 2 L 1
that is, an infinitesimal version of Estimate [START_REF] Belgacem | The effect of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment[END_REF], and one might wonder whether such infinitesimal estimates might lead to global qualitative inequality of the type [START_REF] Belgacem | The effect of dispersal along environmental gradients on the dynamics of populations in heterogeneous environment[END_REF]. This is customary, in the context of shape derivatives. We need to define the notion of shape derivative before going further: Definition 2. Let F : E → F(E) ∈ IR be a shape functional. We define

X 1 (V * ) := Φ : B(0; R) → IR 2 , ||Φ|| W 1,∞ 1 , ∀t ∈ (-1; 1) , χ (Id+tΦ)(B * ) ∈ M(Ω).
as the set of admissible perturbations at E * . The shape derivative of first (resp. second) order of a shape function F at V * in the direction Φ is

F (E * )[Φ] = lim t→0 F (Id + tΦ)E * -F(E * ) t (resp.F (E * )[Φ, Φ] := lim t 2 →0 F (Id + tΦ)E * -F(E * ) -F (E * )(Φ) t 2 .) (7)
A customary way to derive quantitative inequality is to show that, at a given shape E, there holds

F (E)[Φ] = 0 , F (E)[Φ, Φ] > 0
and to lift the last inequality to a quantitative inequality of the form

F (E)[Φ, Φ] C||Φ|| 2 s
where || • || s is a suitable norm; we refer to [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF] for more details but for instance one might have || Φ| ∂E || 2 L 1 which often turns out to be the suitable exponent for a quantitative inequality. This quantitative inequality for shape deformations is usually not enough, and we refer to Section 1.4.1 for more details and bibliographical references.

A remark on the proof

The main innovation of this paper is the proof of Theorem 1 which, although it uses shape derivatives as is customary while proving quantitative inequalities, see Subsection 1.4.1, relies heavily on parametric derivatives. This is allowed by the fact that we are working with a potential defined on the interior of the domain. Furthermore, we also prove that, unlike classical shape optimization, our coercivity norm for the second order shape derivative is the L 2 norm.

Bibliographical references

Quantitative spectral inequalities

Spectral deficit for Faber-Krahn type inequalities: Quantitative spectral inequalities have received a lot of attention for a few decades, and are usually set in a context which is more general than the one introduced here. The main goal of such inequalities were to derive quantitative versions of the Faber-Krahn inequality: for a given parameter β ∈ (0; +∞] and a bounded domain Ω ⊂ IR n , consider the first eigenvalue η β (Ω) of the Laplacian with Robin boundary conditions:

-∆u β,Ω = η β (Ω)u β,Ω in Ω, ∂u β,Ω ∂ν + βu β,Ω = 0 on ∂Ω, (8)
with the convention that η ∞ (Ω) is the first Dirichlet eigenvalue of the Laplace operator. Although it has been known since the independent works of Faber [START_REF] Faber | dass unter allen homogenen membranen von gleicher fläche und gleicher spannung die kreisförmige den tiefsten grundton gibt[END_REF] and Krahn [37] that, whenever Ω * is a ball with the same volume as Ω, there holds

η ∞ (Ω) η ∞ (Ω * ),
the question of providing a sharp lower bound for the so-called spectral deficit

R ∞ (Ω) = η ∞ (Ω) -η ∞ (Ω * )
remained largely open until Nadriashvili and Hansen [START_REF] Hansen | Isoperimetric inequalities in potential theory[END_REF] and Melas, [START_REF] Melas | The stability of some eigenvalue estimates[END_REF], using Bonnesen type inequalities, obtained a lower bound on the spectral deficit involving quantities related to the geometry of domain Ω through the inradius. In a later work, Brasco, De Philippis and Velichkov, [START_REF] Brasco | Faber-krahn inequalities in sharp quantitative form[END_REF], the sharp version of the quantitative inequality, namely:

R ∞ (Ω) CA(Ω) 2 . ( 9
)
Their proof relies uses as a first step a second order shape derivative argument, a series of reduction to small asymmetry regime and, finally, a quite delicate selection principle. We comment in the next paragraph on the role of second order shape derivative for generic quantitative inequalities and the part it plays in our proof. For a survey of the history and proofs of quantitative Faber-Krahn inequalities, we refer to the survey [START_REF] Brasco | Shape optimization and spectral theory[END_REF] and the references therein.

For quantitative versions of spectral inequalities with general Robin boundary conditions, the Bossel-Daners inequality, first derived by Bossel in dimension 2 in [START_REF] Bossel | Membranes élastiquement liées inhomogènes ou sur une surface: Une nouvelle extension du théorème isopérimétrique de rayleigh-faber-krahn[END_REF] and later extended by Daners in all dimensions in [START_REF] Daners | A faber-krahn inequality for robin problems in any space dimension[END_REF] reads:

R β (Ω) := η β (Ω) -η β (B) 0 , β > 0,
and a quantitative version of the Inequality was proved by Bucur, Ferone, Nitsch and Trombetti in [START_REF] Bucur | The quantitative faber-krahn inequality for the robin laplacian[END_REF]:

R β (Ω) CA(Ω) 2 .
Their method for β < ∞ is different from the case of Dirichlet eigenvalue and relies on a free boundary approach.

Quantitative estimates for optimal potentials In the parametric context, that is, when optimising a criterion with respect to a potential, two references whose results are related to the one of the present paper are [START_REF] Brasco | Improved energy bounds for schrödinger operators[END_REF] and [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a schrödinger operator[END_REF]; in both these papers, the L ∞ constraint 0 V 1 we consider in the present paper is not considered and they mainly deal with L p constraints. Namely, in [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a schrödinger operator[END_REF], the main result, in the two dimensional case, is the following: consider, for a parameter γ > 0, a non-postivie potential V ∈ L 1+γ (IR 2 ) , V 0, the operator

L V := -∆ + V
and its first eigenvalue

E(V ) := inf IR 2 |∇ψ| 2 + IR 2 V ψ 2 , ψ ∈ W 1,2 (IR 2 ) , IR 2 ψ 2 = 1 .
The relevant optimisation problem is then

C γ := sup      |E(V )| IR 2 V γ+ 1 2 1 γ     
, and the optimal class is

M γ :=      V , |E(V )| IR 2 V γ+ 1 2 1 γ = C γ , V 0      .
The first result of [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a schrödinger operator[END_REF] is that M γ is non-empty. Furthermore they obtain many stability estimates [16, Theorem 2.2], one of which reads: there exists c γ > 0 such that, for any γ > 0 such that γ

3 2 , for any V ∈ L γ+ 1 2 , V 0, |E(V )| IR 2 V γ+1 1 γ C γ -c γ inf W ∈Mγ V -W γ+1 V 2 γ+1 . ( 10
)
Their proof is strongly related to the stability of functional inequalities, such as the Gagliardo-Nirenberg-Sobolev or Hölder inequalities, and it is not clear to us that their methods can be used in the context we are presently considering.

In [START_REF] Brasco | Improved energy bounds for schrödinger operators[END_REF], a stability estimate for the Dirichlet energy with respect to the potential is obtained. One of their main theorems reads as follows [10, Theorem B]: let Ω be a smooth domain in IR n and f ∈ W -1,2 (Ω). Let, for any potential V enjoying some suitable integrability properties,

u V ∈ W 1,2 0 (Ω) be the solution of -∆u V + V u V = f
and define the associated Dirichlet energy

E f (V ) := - 1 2 Ω |∇u V | 2 - 1 2 Ω V u 2 V .
Define, for any p ∈ (1; ∞),

V p := {V ∈ L p (Ω), V L p 1} .
In [START_REF] Buttazzo | Optimal potentials for schrödinger operators[END_REF], it was proved that a solution V 0 to the maximisation problem sup

V ∈Vp E f (V )
exists, is unique and satisfies V 0 L p = 1. Then [10, Theorem B] reads: there exists

σ p > 0 such that ∀V ∈ V p , E f (V ) E f (V 0 ) -σ p ||V -V 0 || 2 p .
Here, the proof of this stability results relies on stability estimates for functional inequalities, and the optimality conditions are quite different from ours, as the optimality system in their case is a non-autonomous, semilinear equation. It is not clear to us that the constraints 0 V 1 we consider here, as well as the spectral quantity we are optimising, can be handled through the methods of [START_REF] Brasco | Improved energy bounds for schrödinger operators[END_REF].

The role of second order shape derivatives: We only want to mention here the results we draw our inspiration from in Step 4 of the proof, and do not aim at giving out the rigorous mathematical setting of the results mentioned below. We refer to [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF] for a thorough presentation of the link between second order shape derivatives, local shape stability and local quantitative inequalities.

As we said in the previous paragraph, most proofs of quantitative inequalities start with a local quantitative inequality for shape perturbation of the optimum E 1 : namely, if F is a regular enough shape functional and E 1 is an admissible set such that

F(E 1 )[Φ] = 0 , F (E 1 )[Φ, Φ] > 0 (11)
for any Φ ∈ X 1 (E 1 ) then it is proved in [START_REF] Dambrine | On variations of the shape hessian and sufficient conditions for the stability of critical shapes[END_REF], [START_REF] Dambrine | About stability of equilibrium shapes[END_REF] that E 1 is a strict local minimizer in a C 2,α neighbourhood of E (actually, in these two articles, the authors assume a coercivity of the second order derivative in in H 1 2 norm on Φ). In [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF], Dambrine and Lamboley proved that the same conditions imply a local quantitative inequality under certain technical assumptions. Roughly speaking, their result implies the following result: Condition [START_REF] Brasco | Shape optimization and spectral theory[END_REF] implies that if, for any function h ∈ C 0 (∂Ω) ∩ W 2,∞ (∂Ω) we define E h 1 as the domain bounded by

∂E h 1 := {x + h(x)ν E1 (x) , x ∈ ∂E 1 }
then there exists > 0 such that, for any ||h|| W 2,∞ (∂Ω) there holds

F(E 1 ) + C||h|| 2 L 1 F(E h 1 )
for some C > 0.

Their result actually holds in stronger norm, but this is the version we wanted to mention here since it is the one we will adapt in our parametric setting. We note however, that the authors, in their problems, prove their inequalities for shape using a H 1 2 coercivity norm for the second derivative: they usually have

F(E 1 ) [h, h] C||h|| 2 H 1 2
.

In this expression, we have defined

F (E 1 )[h, h] := F(E 1 ) [x + hν E1 (x), x + hν E1 (x)].
Here, in

Step 4, we will show, using comparison principles, that the optimal coercivity norm is the L 2 norm.

Difference with our proofs and contribution: As mentioned in the previous paragraphs, most of the existing literature deals with stability estimates in the context of shape optimisation or, when dealing with optimisation with respect to the potential, with L p , 1 < p < ∞ constraints. We believe this article to be a first step in the context of L ∞ constraints. In this setting, other types of phenomenons appear. As we will see while proving Theorem 1, a shape derivative approach can not be sufficient in of its own for our purposes, as is usually the case, but is needed. Indeed, changes in the topology of competitors may occur, which calls for a new specific method. To tackle the second order shape derivative, we use a comparison principle.

Mathematical biology

We briefly sketch some of the biological motivations for the problem under scrutiny here. Following the works of Fisher, [START_REF] Fisher | The wave of advances of advantageous genes[END_REF], Kolmogoroff, Petrovsky and Piscounoff [START_REF] Kolmogoroff | études de l'équation avec croissance de la quantité de matière et son application à un problème biologique[END_REF], a popular model for population dynamics in a bounded domain is the following so-called logistic-diffusive equation:

   ∂u ∂t = ∆u + u(m -u) in Ω , u = 0 on ∂Ω , u(t = 0) = u 0 0 , u 0 = 0. (12)
In this equation, m ∈ L ∞ (Ω) accounts for the spatial heterogeneity and can be interpreted in terms of resources distribution: the zones {m 0} are favorable to the growth of the population, while the zones {m 0} are detrimental to this population. The particular structure of the nonlinearity -u 2 (which accounts for the Malthusian growth of the population) makes it so that two linear steady states equations are relevant to our study: the steady-logistic diffusive equation

   ∆θ + θ(m -θ) = 0 in Ω , θ = 0 on ∂Ω , θ 0, (13)
and the first eigenvalue equation of the linearization of (12) around the solution z ≡ 0:

-∆ϕ m -mϕ m = λ(m)ϕ in Ω , ϕ = 0 on ∂Ω, (14)
where λ(m) is the first eigenvalue of the operator L m defined in [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF]. More precisely, it is known (see [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF][START_REF] Cantrell | Diffusive logistic equations with indefinite weights: Population models in disrupted environments II[END_REF][START_REF] Skellam | Random dispersal in theoretical populations[END_REF]) that 1. Whenever λ(m) < 0, has a unique solution θ m , and any solution u = u(t, x) of ( 12) with initial datum u 0 0 , u 0 = 0 converges in any L p to θ m as t → ∞.

2. Whenever λ(m) 0, any solution u = u(t, x) of ( 12) with initial datum u 0 0 , u 0 = 0 converges in any L p to 0 as t → ∞.

The eigenvalue which we seek to minimize can thus be interpreted as a measure of the survival ability given by a resources distribution, and later works investigated the problem of minimizing λ(m) with respect to m under the constraint m ∈ M(Ω), where M(Ω) is defined in [START_REF] Alvino | A lower bound for the first eigenvalue of an elliptic operator[END_REF]. In other words, this is the problem (5). In the case of Neumann boundary conditions, Berestycki, Hamel and Roques introduced the use of a rearrangement (due to Berestycki and Lachand-Robert, [START_REF] Berestycki | Some properties of monotone rearrangement with applications to elliptic equations in cylinders[END_REF]) in that context, see [START_REF] Berestycki | Analysis of the periodically fragmented environment model : I -species persistence[END_REF] and [START_REF] Kawohl | Rearrangements and Convexity of Level Sets in PDE[END_REF] for an introduction to rearrangement, and further geometrical properties of optimizers were derived by Lou and Yanagida, [START_REF] Lou | Minimization of the principal eigenvalue for an elliptic boundary value problem with indefinite weight, and applications to population dynamics[END_REF], by Kao, Lou and Yanagida [START_REF] Kao | Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[END_REF]. We do not wish to be exhaustive regarding the literature of this domain and refer to [START_REF] Lamboley | Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions[END_REF] where Lamboley, Laurain, Nadin and Privat investigate several properties of solutions of (5) under a variety of boundary conditions, and the references therein.

Proof of Theorem 1 2.1 Background on (5) and structure of the proof

We recall that we work in Ω = B(0; R) ⊂ IR 2 , that

M := M B(0; R) = 0 V 1 , Ω V = V 0
and that r * is chosen so that

|B(0; r * )| = V 0 i.e such that V * = χ B(0;r * ) ∈ M.
We define S * := ∂B * .

We first recall the following simple consequence of Schwarz' rearrangement:

Lemma 1. V * is the unique minimizer of λ in M: for any V ∈ M, λ(V * ) < λ(V ).
The associated eigenfunction u * is decreasing and radially symmetric.

This result is well-known, but for the sake of completeness we prove it in Annex A. The proof of Theorem 1 relies on the study of two auxilliary problem: we introduce, for a given δ > 0, the new admissible sets

M δ := {V ∈ M , ||V -V * || L 1 = δ} , (15)
Mδ := {V radially symmetric, V ∈ M , ||V -V * || L 1 = δ} (16)
and study the two variational problems inf

V ∈ Mδ λ(V ) (17)
and inf

V ∈M δ λ(V ). (18)
Obviously, Theorem 1 is equivalent to the existence of C > 0 such that

∀δ > 0 , ∀V ∈ M δ , λ(V ) -λ(V * ) Cδ 2 . ( 19
)
Remark 2. This is a parametric version of the selection principle of [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF], that was developed in [START_REF] Brasco | Faber-krahn inequalities in sharp quantitative form[END_REF]. We refer to [START_REF] Brasco | Shape optimization and spectral theory[END_REF] for a synthetic presentation of this selection principle. We note however that the fact that they use a perimeter constraint enables them to prove that a solution to their auxiliary problem is a normal deformation of the optimal shape. The main difficulty in the analysis of [START_REF] Brasco | Faber-krahn inequalities in sharp quantitative form[END_REF] is establishing C 2 bounds for this normal deformation. Here, working with subsets as shape variables gives, from elliptic regularity, enough regularity to carry out this step when the solution of the auxiliary problem is a normal deformation of B * . However, we conjecture that the solutions of ( 17) and ( 18) are equal and are disconnected (see Step 3 and the Conclusion for a precise conjecture), so that the core difficulty is proving that handling the inequality for normal deformations and for radial distributions is enough to get the inequality for all other sets.

To prove [START_REF] Cosner | Does movement toward better environments always benefit a population?[END_REF] we follow the steps below:

1. We first show that ( 18) and ( 17) have solutions. The solutions of ( 18) will be denoted by V δ , the solution of ( 17) will be denoted by H δ .

2. We prove that it suffices to establish [START_REF] Cosner | Does movement toward better environments always benefit a population?[END_REF] for δ small enough.

3. For [START_REF] Caubet | Optimal location of resources for biased movement of species: the 1D case[END_REF] we fully characterize the solutions for δ > 0 small enough and prove that

∀δ > 0 , ∀V ∈ Mδ , λ(V ) -λ(V * ) Cδ 2 . ( 20
)
In other words, we prove that Theorem 1 holds for radially symmetric functions.

4. We compute the first and second order shape derivatives of the associated Lagrangian at the ball B * and prove a L 2 -coercivity estimate for the second order derivative. We comment upon the fact that (unlike many shape optimization problems) this is the optimal coercivity norm at the beginning of this Step. We use this information to prove that Theorem 1 holds for domains that are small normal deformations of B * with bounded mean curvature.

5. We establish a dichotomy for the behaviour of V δ and prove that ( 19) holds for any V by using (20) and Step 4.

Step 1: Existence of solutions to (17)-(18)

We prove the following Lemma:

Lemma 2. The optimization problems (17) and (18) have solutions.

Proof of Lemma 2. The proof follows from the following claim:

Claim 1. M δ and Mδ are compact for the weak L ∞ topology.

We postpone the proof to the end of this Proof. Lemma 2 follows from this claim, and we only write the details for [START_REF] Cianchi | A quantitative polya?szego principle[END_REF]. Let {V k } k∈IN be a minimizing sequence for λ in M δ . From Claim 1, there exists

V ∞ ∈ M δ such that V k V ∞ .
The notation stands for the weak convergence in the weak L ∞ -* sense. Let, for any k ∈ IN,

u k := u V k and λ k := λ(V k ).
We first note that the sequence {λ k } k∈IN is bounded. Indeed, let ϕ ∈ W 1,2 0 (Ω) be such that Ω ϕ 2 = 1. From the formulation in terms of Rayleigh quotients and V k 0 there holds

λ k Ω |∇ϕ| 2 - Ω V k ϕ 2 Ω |∇ϕ| 2 .
This gives an upper bound. For a lower bound, let λ 1 (Ω) be the first Dirichlet eigenvalue of Ω (equivalently, this is the eigenvalue associated with V = 0). From V 1, Ω u 2 k = 1 and the variational formulation for λ 1 (Ω) there holds

λ 1 (Ω) -1 Ω |∇u k | 2 - Ω V u 2 k λ k
so that the sequence also admits a lower bound. It is straightforward to see that {u k } k∈IN is bounded in W 1,2 0 (Ω) so that from the Rellich-Kondrachov Theorem there exists u ∞ ∈ W 1,2 0 (Ω) such that {u k } k∈IN converges strongly in L 2 and weakly in W 1,2 0 (Ω) to u ∞ . Passing to the limit in the weak formulation

∀v ∈ W 1,2 0 (Ω) , Ω ∇u k , ∇v - Ω V k u k v = λ(V k ) Ω u k v, in the normalization condition Ω u 2 k = 1
and in u k 0 readily shows that u ∞ is a non-trivial eigenfunction of L V∞ . Furthermore, it is non-negative. Since the first eigenfunction is the only eigenfunction with a constant sign, this proves that λ ∞ = λ(V ∞ ) and that u ∞ is the eigenfunction associated with V ∞ . Thus:

λ(V ∞ ) = inf V ∈M δ λ(V ).

It remains to prove Claim 1:

Proof of Claim 1. We only prove it for

M δ . Let {V k } k∈IN ∈ M IN δ . We define, for any k ∈ IN h k := V k -V * .
Since V * = χ B * and 0 V k 1, the following signe conditions hold on h k :

h k 0 in (B * ) c , h k 0 in B * . ( 21
)
Since Ω V k = Ω V * there holds

B * h k = - (B * ) c h k . (22)
Finally from

||V k -V * || L 1 = δ there comes δ = Ω |V k -V * | (23) = Ω |h k | (24)
=

(B * ) c h k - B * h k from (21) (25)
= -2

B * h k from (22) (26)
= 2

(B * ) c h k . ( 27
)
We see 27) and ( 21) we have

h k χ (B * ) c as an element of L ∞ ((B * ) c ). Let h + ∞ be a weak-L ∞ closure point of {h k } k∈IN in L ∞ ((B * ) c ). From (
1 h + ∞ 0 , (B * ) c h + ∞ = δ 2 . ( 28
)
For the same reason, there exists

h - ∞ ∈ L ∞ (B * ) such that -1 h - ∞ 0 , B * h - ∞ = - δ 2 and h k h ∞ weak- * in L ∞ (B * ). ( 29
)
We define

h ∞ := h - ∞ χ (B * ) c + h - ∞ χ B * and it is clear that h k h ∞ weak- * in L ∞ (Ω). Setting V ∞ := V * + h ∞ there holds V k V ∞
and, by ( 28),( 29), V ∞ ∈ M δ (Ω).

2.3

Step 2: Reduction to small L 1 neighbourhoods of V *

We now prove the following Lemma:

Lemma 3. To prove Theorem 1, it suffices to prove [START_REF] Cosner | Does movement toward better environments always benefit a population?[END_REF] for δ small enough, in other words it suffices to prove that there exists C > 0 such that

lim inf δ→0 inf V ∈M δ λ(V ) -λ(V * ) δ 2 C.
Proof of Lemma 3. We define, for any

V = V * , G(V ) := λ(V ) -λ(V * ) ||V -V * || 2 L 1
and consider a minimizing sequence {V k } k∈IN for G. Then we either have, up to subsequence,

∀k ∈ IN , ||V k -V * || L 1 ε > 0 or ||V k -V * || L 1 → k→∞ 0.
In the first case, up to a converging subsequence, V k V ∞ in a weak L ∞ - * sense and, by the same arguments as in the Proof of Claim 1,

||V ∞ -V * || L 1 ε > 0.
Furthermore, by the same arguments as in the proof of Lemma 2,

λ(V k ) -→ k→∞ λ(V ∞ ) so that G(V k ) -→ k→∞ λ(V ∞ ) -λ(V * ) ||V ∞ -V * || L 1 and, by Lemma 1, G(V ∞ ) > 0. Hence we only need to study the case ||V k -V * || L 1 → k→∞ 0, as claimed.
The same arguments yield the following Lemma: Lemma 4. (20) is equivalent to proving that there exists C > 0 such that

lim inf δ→0 inf V ∈ Mδ λ(V ) -λ(V * ) δ 2 C.

2.4

Step 3: Proof of (20)

In this Subsection we prove [START_REF] Dambrine | On variations of the shape hessian and sufficient conditions for the stability of critical shapes[END_REF] or, in other words, we prove that Theorem 1 holds for radial distributions.

Proof of [START_REF] Dambrine | On variations of the shape hessian and sufficient conditions for the stability of critical shapes[END_REF]. We recall that, by Lemma 2, there exists a solution to [START_REF] Caubet | Optimal location of resources for biased movement of species: the 1D case[END_REF]. Let H δ be such a minimizer. We first characterize H δ for δ small enough. Let

A δ := {|x| r * + r δ } \ {x , r * -r δ |x| r * + r δ } (30)
be the annular structure such that

χ A δ ∈ Mδ . (31)
We represent it below

A δ A δ Figure 1: An example of A δ
Since r δ is defined by the relation

|{r * |x| r * + r δ }| = δ 2
the set A δ is uniquely defined. We claim the following:

Claim 2. There exists δ > 0 such that, for any δ δ,

H δ = χ A δ .
Proof of Claim 2. To prove this claim, we need the optimality conditions associated with [START_REF] Caubet | Optimal location of resources for biased movement of species: the 1D case[END_REF]. We first note that, if u δ is the eigenfunction associated with H δ , there exist two real numbers η δ , µ δ such that

1. H δ = χ {u δ >µ δ }∩B * in B * , 2. H δ = χ {sup S * u δ u δ >η δ }∩(B * ) c in (B * ) c , 3. |{u δ > µ δ } ∩ B * | = V 0 -δ 2 , |{sup S * u δ u δ > η δ } ∩ (B * ) c | = δ 2
. This is readily seen from the Rayleigh quotient formulation (3). We only prove 1: let µ δ ∈ IR be the only real number such that

|{u δ > µ δ } ∩ B * | = V 0 - δ 2 = B * H δ and replace H δ by Hδ := χ {u δ >µ δ }∩B * + H δ χ (B * ) c .
Since u δ is radially symmetric (because H δ is radially symmetric), Hδ is radially symmetric. Then, because B * Hδ = B * H δ , we have, by the bathtub principle (see [START_REF] Henrot | Shape Variation and Optimization[END_REF])

B * u 2 δ H δ B * u 2 δ χ {u δ >µ δ }∩B * , hence λ(H δ ) B |∇u δ | 2 - B Hδ u 2 δ λ Hδ .
This gives the required property. We now need to exploit these optimality conditions. First of all, since u δ is radially symmetric, we can define

ζ δ := u δ | S * .
By standard elliptic estimates, we also have

u δ → δ→0 u * in C 1,s ( s < 1),
where u * is the eigenfunction associated with V * . Since ∂u * ∂r < -c on {|x| ε}, u δ is radially decreasing in {|x| > ε} for δ > 0 small enough. It follows that µ δ > ζ δ for δ small enough. For the same reason, ζ δ > η δ for δ small enough. Hence we have

µ δ > η δ > η δ , H δ = χ {u δ >µ δ } + χ {ζ δ u δ >η δ } .
Finally, once again because u δ is radially decreasing on {|x| > ε} for δ small enough, both level sets {u δ > µ δ } and {ζ δ u δ > η δ } are connected, and H δ is the characteristic function of a centered ball and of an annulus, i.e

H δ = χ { x r * -z δ } + χ {r * x r * +y δ } . Since |A δ ∆B * | = B |H δ -V * |
there holds

H δ = χ A δ
for δ small enough, as claimed.

We now turn to the proof of ( 20): since H δ is the minimizer of λ in Mδ we are going to prove that there exists a constant C > 0 such that

λ(H δ ) λ(V * ) + Cδ 2 (32)
for δ δ. Because of Lemma 3, (20) will follow. To prove [START_REF] Henrot | Shape optimization of the dirichlet energy for semilinear elliptic partial differential equations[END_REF], we use parametric derivatives. Let us fix notations:

1. For any δ > 0 small enough so that H δ = χ A δ , we set

h δ := H δ -V * .
2. For any t ∈ [0; 1] we define V δ,t as V δ,t := V * + th δ and u δ,t as the eigenfunction associated with V δ,t :

   -∆u δ,t -V δ,t u δ,t = λ t u δ,t in Ω, u δ,t = 0 on ∂Ω. Ω u 2 δ,t = 1. ( 33
)
3. For any such δ, uδ is the first order parametric derivative of u at V δ,t in the direction h δ and λ δ,t is the first order parametric derivative of λ at V δ,t in the direction h δ .

As is proved in Annex B, these objects are well-defined. Differentiating the equation with respect to t gives

   -∆ uδ,t = λ(V * ) uδ,t + V * uδ,t + λδ,t u * + h δ,t u * , in Ω, uδ,t = 0 on ∂Ω, B u * uδ,t = 0, (34)
and , multiplying the first equation by u δ,t and integrating by parts ,

λδ,t = - B h δ u 2 δ,t .
We apply the mean value Theorem to f : t → λ δ,t . This gives the existence of

t 1 ∈ [0; 1] such that λ(H δ ) -λ(B * ) = f (1) -f (0) = f (t 1 ) = - B h δ u 2 δ,t1 .
Our goal is now to prove that

- B h δ u 2 δ,t1 Cδ 2 (35)
for some constant C > 0 whenever δ is small enough. We will actually prove the existence of δ > 0 such that, for any t ∈ [0; 1] and any δ δ, there holds

- B h δ u 2 δ,t Cδ 2 (36)
for some C > 0.

Proof of Estimate [START_REF] Kolmogoroff | études de l'équation avec croissance de la quantité de matière et son application à un problème biologique[END_REF]. We recall that r δ and r δ were defined in ( 30)- [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF]. We can rewrite

H δ = χ A δ ∈ X δ under the form r * r * -r δ t n-1 dt + r * +r δ r * t n-1 dt = δ c n
where c n = H n-1 (S(0; 1)) and the condition Ω h δ = 0 implies

r * r * -r δ t n-1 dt = r * +r δ r * t n-1 dt.
An explicit computation yields the existence of a constant C > 0 such that

r δ , r δ ∼ δ→0 Cδ. (37)
Let

I ± δ := {h δ = ±1}. Since h δ is radial, for any t ∈ [0; 1] the function u δ,t is radial.
First facts regarding u δ,t Identifying u δ,t (resp. V δ,t ) with. the unidimensional function ũδ,t (resp. Ṽδ,t ) such that u δ,t (x) = ũδ,t (|x|) (resp. V δ,t (x) = Ṽδ,t (|x|))

we have the following equation on u δ,t :

     -1 r n-1 r n-1 u δ,t = V δ,t u δ,t + λ t u δ,t in [0; R], u δ,t (0) = u δ,t (1) = 0, 1 0 xu δ,t (x) 2 dx = 1 cn . (38)
Since V δ,t is constant in (r * -r δ ; r * ) ∪ (r * ; r * + r δ ) and since u δ,t is uniformly bounded in L ∞ by standard elliptic estimates, u δ,t is C 2 in (r * -r δ ; r * ) ∪ (r * ; r * + r δ ). Furthermore, Equation [START_REF] Lamboley | Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions[END_REF] readily gives the existence of a constant M such that, uniformly in δ and in t ∈ [0; 1],

||u δ,t || W 2,∞ M. (39)
Finally, it is standard to see that Equation [START_REF] Henrot | Shape Variation and Optimization[END_REF] gives

||u δ,t -u * || C 1 → δ→0 0 (40)
uniformly in t. As a consequence, since u * (r * ) < 0 there exists δ 1 > 0 such that, for any δ δ 1 and any t ∈ [0; 1],

u δ,t (r * ) -C < 0. ( 41
)
End of the Proof For any x ∈ I ± δ and any t ∈ [0; 1], a Taylor expansion gives

u 2 δ,t (x) = u 2 δ,t
I + δ dist(x; S * ) .
However,

I + δ dist(x; S * ) = r * +r δ r * t n-1 (t -r * )dt = (r * ) n r δ 1 n + 1 n + 1 n - 1 n n n -1 + (r * ) n-1 (r δ ) 2 1 n + 1 n + 1 n -1 - 1 n n n -2 + o δ→0 (δ 2 ) by (37) = (r * ) n-1 (r δ ) 2 + o δ→0 (δ 2 ) ∼ δ→0 Cδ 2
for some C > 0 by [START_REF] Krahn | Über eine von rayleigh formulierte minimaleigenschaft des kreises[END_REF]. In the same manner,

I - δ dist(x; S * ) = r * r * -r δ t n-1 (r * -t)dt ∼ δ→0 C δ 2
and so, combining these estimates gives λδ,t

δ→0 Cδ 2 + o(δ 2 )
uniformly in δ and t, which concludes the proof.

2.5

Step 4: shape derivatives and quantitative inequality for graphs

Preliminaries and notations

In this Subsection, we aim at proving Theorem 1 for V = χ E and where E can be obtained as a normal graph over B * .

Introduction of the Lagrangian and optimality conditions

We introduce the Lagrange multiplier τ associated with the volume constraint and define the Lagrangian

L τ : E → λ(E) -τ V ol(E).
From classical results in the calculus of variations, we have the following optimality conditions. Claim 3. The necessary optimality conditions for a shape E to be a local minimizer (however they are not sufficient) are:

∀Φ ∈ X 1 (E) , λ (E)[Φ] = 0 , ∀Φ ∈ X 1 (E) , L τ (E)[Φ, Φ] > 0.
Here, we use the notion of shape derivatives introduced in Definition 2. Since we only want a local quantitative inequality for shapes that can be obtained as normal graphs over the ball B * , we introduce some notations.

Notations We consider in this parts functions g belonging to

X 0 (B * ) = g ∈ W 1,∞ (B * ) , ||g|| L ∞ (∂B * ) 1 , ∂B * g = 0 .
Whenever g ∈ X 0 (B * ), there exists Φ g ∈ X 1 (B * ) such that Φ g , ν = g on ∂B * .

The set X 0 corresponds to a linearization of the volume constraints for normal graphs and can be seen as a subset of the set of admissible perturbations X 1 (B * ) defined in Definition 2, in the sense that we restrict admissible perturbations to normal graphs.

g(x)

Figure 2: A normal deformation of the ball. The dotted line can be understood as the graph of g. We define, for any g ∈ X 0 (B * ) and any t ∈ [0; T ] (with T uniform in g because of the L ∞ constraint) the set B t,g whose boundary is defined as

∂B t,g := {x + tg(x)ν(x) , x ∈ ∂B * }
i.e a slight deformation of E * . We define λ g,t := λ (B t,g ) .

Recall that τ is the Lagrange multiplier associated with the volume constraint. By defining

L τ (B * )[g] := L τ (B * )[Φ g ] and L τ (B * )[g, g] := L τ (B * )[Φ g , Φ g ],
and with the same convention for other shape functionals involved, necessary optimality conditions are

∀g ∈ X 0 (B * ) , L τ (B * )[g] = 0 , L τ (B * )[g, g] > 0. ( 42
)
We first prove that these optimality conditions hold in the case of a ball and then use them to obtain Theorem 1 for normal perturbations.

Strategy of proof and comment on the coercivity norm

The strategy of proof is the same as the one used in many articles devoted to quantitative spectral inequalities. For example, we refer to [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF][START_REF] Brasco | Faber-krahn inequalities in sharp quantitative form[END_REF] for applications of these methods and to the recent [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF], which presents a general framework for the study of stability and local quantitative inequalities using second order shape variations.

Although our method of proof is similar, we point out that the main thing to be careful with here is the coercivity norm for the second-order shape derivative. Indeed, let J : E → J(E) be a differentiable shape function. In the context of shape spectral optimization, the "typical" coercivity norm at a local minimum E * for J is the H 1 2 norm: in [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF] a summary of shape functionals known to satisfy

J (E * )[Φ, Φ] C|| Φ, ν || 2 H 1 2 (∂E * )
is established and proofs of thess coercivity properties are given. In this estimate, Φ is an admissible vector field at E * .

Here, in the context of parametric shape derivatives, i.e when the shape is a subdomain, it appears (see Subsubsection 2.5.5) that the natural coercivity norm is the L 2 norm:

L τ (E * )[Φ, Φ] C|| Φ, ν || 2 L 2 (∂A * )
and this coercivity norm is optimal. This makes things a bit more complicated when dealing with the terms of the second order derivative that involve the mean curvature. This lack of coercivity might be accounted for by the fact that, while in shape optimization, it is the normal derivative of the shape derivative of the eigenfunction that is involved (see [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF]) here, it is just the trace of the shape derivative of the eigenfunction on the boundary of the optimal shape that matters. Once this L 2 coercivity is established, we will prove that there exists a constant ξ, M, C > 0 such that, for any g satisfying ||g|| W 1,∞ ξ and such that the mean curvature of B g,t is bounded by M for any t T , there holds

|L τ (B t,g )[Φ g , Φ g ] -L τ (B * * )[Φ g , Φ g ]| (C + M )||g|| W 1,∞ ||g|| 2 L 2 . ( 43
)
We then apply the Taylor-Lagrange formula to f : t → L τ (B t,g ) to get the desired conclusion, see Subsubsection 2.5.7.

Analysis of the first order shape derivative at the ball and computation of the Lagrange multiplier

The aim of this section is to prove the following Lemma:

Lemma 5. B * is a critical shape and the Lagrange multiplier associated with the volume constraint is

τ = -u 2 * | ∂B * .
Proof of Lemma 5. We recall (see [START_REF] Henrot | Shape Variation and Optimization[END_REF]) that

V ol (B * )[g] = ∂B * g (= 0 if g ∈ X 0 (B * )) .
We now compute the first order shape derivative of λ. The shape differentiability of λ follows from an application of the implicit function Theorem of Mignot, Murat and Puel, [START_REF] Mignot | Variation d'un point de retournement par rapport au domaine[END_REF], and is proved in Appendix B. Let, for any g ∈ X 0 (B * ), u g be the shape derivative of the u Bt,g at t = 0. We recall that this derivative is defined as follows (see [START_REF] Henrot | Shape Variation and Optimization[END_REF]Chapitre 5] for more details): we first define v t := u g,t •Φ g,t , we define ug as the derivative in W 1,2 0 (Ω) of v t with respect to t at t = 0, and set u g := ug -Φ g , ∇u 0 .

We proceed formally to get the equation on u g (for rigorous computations we refer to Appendix B): we first differentiate the main equation -∆u g,t = λ g,t u g,t + V g,t u g,t with respect to t, yielding -∆u g = λ g u * + λ * u g + (V * )u g .

We then differentiate the continuity equations to get the jump conditions: if we define

[f ](x) := lim y→x,y∈(B * ) c f (y) -lim y→x,y∈B * f (y),
we have

[u g,t ]| ∂Bt,g = ∂u g,t ∂ν ∂Bt,g = 0, yielding u g ∂B * = -g ∂u * ∂ν ∂B * = 0 because u * is C 1 , and ∂u g ∂ν ∂B * = -g ∂ 2 u * ∂ν 2 ∂B * .
However, from the Equation on u * we see that

∂ 2 u * ∂ν 2 ∂B * = u * | ∂B * ,
so that we finally have the following equation on u g :

-∆u g = λ g u * + λ * u * + (V * )u * in B(0; R), ∂u g ∂ν = -gu * | ∂B * . (44)
The weak formulation of this equation reads: for any ϕ ∈ W 1,2 0 (B),

B ∇u g , ∇ϕ - ∂B * gu * ϕ = λ g B ϕu * + λ * B ϕu * + B (V * )ϕu * . (45)
We finally remark that, by differentiating B u 2 g,t = 1, we get

B u * u g = 0.
Taking u * as a test function in [START_REF] Skellam | Random dispersal in theoretical populations[END_REF] and using the normalization condition B u 2 * = 1 thus gives

λ g = B ∇u * , ∇u g -λ * B u g u * - B (V * )u * u g - ∂B * gu 2 * . (46)
However, since u g does not have a jump at ∂B * , we have

B ∇u * , ∇u g -λ * B u g u * - B (V * )u * u g = 0
by using the Equation on u * .

In the end, we get

λ g = - ∂B * gu 2 * .
Since u 2 * = µ * is constant on ∂B * and g ∈ X 0 (B * ),

λ g = -µ * ∂B * g = 0.
This also enables us to compute the Lagrange multiplier: for a function g ∈ W 1,∞ (∂B * ) which is no longer assumed to satisfy ∂B * g = 0, one must have

L τ [B * ](g) = 0.
Indeed, we know, from Lemma 1, that B * is the unique minimizer of λ under the volume constraint. However, the same computations show that

L τ [B * ](g) = -µ * ∂B * g -τ ∂B * g,
and the Lagrange multiplier is thus

τ = -µ * = -u 2 * | B * . (47)
We now compute the second order shape derivative of L τ at any given shape.

Computation of the second order shape derivative of λ

We explained in Subsection 2.5.2 that we need to compute the second order derivative at any given shape in order to apply the Taylor-Lagrange formula. Thus, the objectif of this section is the proof of the following Lemma: Lemma 6. The second order derivative of the eigenvalue λ at a shape E in the direction Φ ∈ X 1 (E) is given by

λ (E)[Φ, Φ] = -2 ∂E uu Φ, ν +2 ∂E ∂u ∂ν ∇ 2 u[Φ, Φ] - ∂ 2 u ∂ν 2 Φ, ν 2 + ∂E -Hu 2 -2u ∂u ∂ν Φ, ν 2 ,
where u is defined by Equation ( 44) and H is the mean curvature of E.

Proof of Lemma 6. To compute λ (E)[Φ, Φ], we use Hadamard's second variation formula (see [33, Chapitre 5, page 227]): let K be a C 2 domain, f (t) be a shape differentiable function , then

d 2 dt 2 t=0 KΦ,t f (t) = K f (0) + 2 ∂K f (0)g + ∂K Hf (0) + ∂f (0) ∂ν g 2 . ( 48
)
Let u be the shape derivative of u E with respect to t and u the second order shape derivative of u E with respect to t. We successively apply (48) to E Φ,t and

f (t) = |∇u t | 2 -u 2 t and to (E Φ,t ) c and f (t) = |∇u t | 2 . Since λ(E Φ,t ) = EΦ,t |∇u t | 2 -u 2 t + E c Φ,t |∇u t | 2 , this gives λ (E)[Φ, Φ] =2 Ω ∇u , ∇u -2 E u u (49) + 2 Ω |∇u | 2 -2 E (u ) 2 (50) + 4 ∂E [ ∇u, ∇u ] Φ, ν -4 ∂E u u Φ, ν (51)
+ ∂E -Hu 2 -2 ∂u ∂ν ∂ 2 u ∂ν 2 -2u ∂u ∂ν Φ, ν 2 . ( 52
)
Let us simplify this expression: First of all the weak formulation of the equation on u gives

2 Ω |∇u | 2 -2 E (u ) 2 = 2λ Ω u u =0 since Ω u u=0 +2λ Ω (u ) 2 -2 ∂E u ∂u ∂ν .
We also note that by differentiating Ω u 2 t = 1 twice with respect to t we get

Ω u u + Ω (u ) 2 = 0. ( 53
)
We note one last simplification to handle Line (51) in the expression for λ : we decompose

∇u = ∂u ∂ν ν + ∇ ⊥ u , ∇ ⊥ u, ν = 0.
We adopt the same decomposition for u and notice that, since u does not have a jump at ∂E,

∇ ⊥ u = 0.
The notation 0 stands for the zero vector in IR n . The same holds true for u, and so, since ∂u ∂ν has no jump at ∂E, we get [ ∇u, ∇u ] = ∂u ∂ν ∂u ∂ν .

Finally, using the weak formulation of the equation on u (Equation ( 2)) we get

2 Ω ∇u , ∇u -2 E u u = 2λ Ω u u - ∂E [u ] ∂u ∂ν .
We then only need to compute the jump [u ] at ∂E. However, invoking the W 2,2 regularity of the material derivative ü, we get for the shape derivative u :

[u ] = -2 [∇u [Φ]] -[∇u[DΦ(Φ)]] -∇ 2 u [[Φ, Φ]] .
We now use the fact that

[∇u] = ∇ ⊥ u = ∇ ⊥ u = 0 on ∂E to rewrite [u ] = -2 ∂u ∂ν Φ, ν -∇ 2 u [[Φ, Φ]] .
If we gather these expressions we get

λ (E)[Φ, Φ] = 2λ Ω u u + 2λ Ω (u ) 2 (= 0 because of (53)) + 4 ∂E ∂u ∂ν ∂u ∂ν Φ, ν + 2 ∂E ∂u ∂ν ∇ 2 u [[Φ, Φ]] + 2 ∂E u u - 4 ∂E ∂u ∂ν ∂u ∂ν Φ, ν -4 ∂E u u Φ, ν + ∂E -Hu 2 -2 ∂u ∂ν ∂ 2 u ∂ν 2 -2u ∂u ∂ν Φ, ν 2 = -2 ∂E uu Φ, ν + 2 ∂E ∂u ∂ν ∇ 2 u[Φ, Φ] - ∂ 2 u ∂ν 2 Φ, ν 2 + ∂E -Hu 2 -2u ∂u ∂ν Φ, ν 2

Analysis of the second order shape derivative at the ball

The aim of this paragraph is to prove the following Lemma:

Proposition 1. There exists a constant C > 0 such that ∀g ∈ X 0 (B * ) , L τ [B * ](g, g) C||g|| 2 L 2 (∂B * )
. We note that the proof of this Lemma relies on a monotonicity principle, which guarantees the weak L 2 coercivity. In fact, this is to be the optimal coercivity, in sharp contrast with shape optimization with respect to the boundary of the whole domain ∂Ω, where the optimal coercivity usually occurs in the H 1 2 norm, as noted in Subsection 2.5.2.

Proof of Proposition 1. We proceed in several steps. We identify g with the normal vector field Φ g that can be constructed from g, and, to alleviate notations, write Φ = Φ g .

Computation of λ

We use Lemma 6 and first note that, since the vector field Φ associated with g is normal,

∇ 2 u[Φ, Φ] = ∂ 2 u ∂ν 2 Φ, ν 2 .
In the case of a ball, H = 1 r * . For notational simplicity, we stick to the notation

H * = 1 r * .
The second derivative of λ becomes

λ (B * )[Φ, Φ] = ∂B * -H * u 2 * -2u * ∂u * ∂ν Φ, ν 2 -2 ∂B * u * u Φ, ν .
Taking into account the value of the Lagrange multiplier τ associated with the volume constraint, see Equation (47), and

V ol (B * )[Φ, Φ] = ∂B * H * Φ, ν 2 we get L τ (B * )[Φ, Φ] = 2 ∂B * -u * ∂u * ∂ν Φ, ν 2 -u * u Φ, ν . (54)
2. Separation of variables and first simplifications We identify g with a function g : [0; 2π] → IR. We write the decomposition of g as a Fourier series:

g = k∈IN α k cos(k•) + β k sin(k•).
Since g ∈ X 0 (B * ) we have α 0 = 0 and thus

g = ∞ k=1 α k cos(k•) + β k sin(k•). ( 55
)
We define, for any k ∈ IN * , u k (resp. w k ) as the shape derivative of u with respect to the perturbation g = cos(k•) (resp. g = sin(k•)). Since B * is a critical shape from Lemma 5,

u k satisfies    -∆u k = λ * u k + V * u k , [u k ] = -u * cos(k•) on ∂B * , u k = 0 on ∂Ω.
Since u * is constant on partial B * , we can write, in polar coordinates

u k (r, θ) = ψ k (r) cos(kθ)
where ψ k satisfies the following equation (and we identify V * with the one dimensional function

Ṽ * such that (V * )(x) = Ṽ * (|x|) = χ |x| r * ):      -1 r (rψ k ) = λ * + V * -k 2 r 2 ψ k [ψ k ] (r * ) = -u * (r * ) ψ k (R) = 0. (56)
In the same way, we have w k (r, θ) = ψ k (r) sin(kθ).

Whenever g admits the Fourier decomposition (55), the linearity (with respect to g) of the equation on u g gives

u g = ∞ k=1 α k u k + β k w k .
Plugging this in the expression of L τ , see Equation (54), and using the orthogonality prop-

erties of cos(k•), sin(k ) k 1 finally yields L τ (B * )[g, g] = L τ (B * )[Φ, Φ] = ∞ k=1 α 2 k + β 2 k u * | ∂B * - ∂u * ∂ν ∂B * -ψ k (r * ) .
We define the relevant sequence {ω k } k∈IN * :

∀k ∈ IN * , ω k := - ∂u * ∂ν ∂B * -ψ k (r * ), ( 57
) so that L τ (B * )[g, g] = u * | ∂B * ∞ k=1 ω k α 2 k + β 2 k . ( 58
)
Our goal is now the following Lemma:

Lemma 7. There exists C > 0 such that ∀k ∈ IN * , ω k C > 0.
In order to prove this Lemma, we use a comparison principle for one-dimensional differential equations.

3. Proof of Lemma 7: monotonicity principle We will first prove that

∀k ∈ IN * , ω k ω 1 . ( 59
)
We first note that

ψ 1 > 0 in (0; R). ( 60
)
Before we prove, let us see how (60) implies (59). Let, for any k 2, z k be the function defined as

z k := ψ k -ψ 1 .
Since ψ k can be expressed as

ψ k = A k J k ( r R ) for r r * where J k is the k-th Bessel function of first kind, we have z k (0) = z k (R) = 0. Furthermore, z k satisfies - 1 r (rz k ) = λ * + V * - k 2 r 2 ψ k -λ * + V * - 1 r 2 ψ 1 λ * + V * - k 2 r 2 ψ k -λ * + V * - k 2 r 2 ψ 1 because ψ 1 > 0 by (60) λ * + V * - k 2 r 2 z k . ( 61
)
We also have

[z k ](r * ) = 0.
However, this equation and this no-jump condition imply

z k 0. ( 62
)
For k large enough, this simply follows by a contradiction argument: if

λ * + V * -k 2 r 2 < 0 in (0; R) then, if z k reached a positive maximum at some interior point r, we should have 0 - 1 r (rz k (r)) λ * + V * - k 2 r 2 z k (r) < 0, yielding a contradiction.
A proof of (62) that is valid for all values of k reads as follows: identifying u * with its onedimensional counterpart (i.e with the function ũ * :

[0; R] → IR such that u * (x) = ũ * (|x|)) we define p k := z k u * .
We notice that p k (0) = 0 and that

[p k ](r * ) = 0.
Furthermore, by straightforward computation, p k satisfies

- 1 r (rp k ) = -1 u * (rz k ) + z k u 2 * 1 r (ru * ) + 2 u * u * p k .
By (61) and by non-negativity of u * we get

- 1 r (rp k ) - k 2 r 2 p k .
From this it is straightforward to see by a contradiction argument that p k can not reach a positive maximum at an interior point. It remains to exclude the case p k (R) > 0.

We argue, once again, by contradiction, and assume that p k (R) > 0. Since by l'Hospital rules we have

p k (r) ∼ r→R z k (r) u * (r)
we must have z k (r) 0. However once again by l'Hospital's rule,

p k (r) = u * z k u 2 * - u * z k u 2 * ∼ r→0 1 2 z k 2u * < 0.
Hence p k is locally decreasing at R, yielding a contradiction. Thus p k 0 and in turn ψ k -ψ 1 = z k 0, completing the proof of (59). The proof of (60) follows from the same arguments: we define

Ψ 1 := ψ 1 u * and observe that - 1 r (rΨ 1 ) = - 1 r 2 Ψ 1 + 2 u * u * Ψ 1 , [Ψ 1 ](r * ) = -u * (r * ).
We once again argue by contradiction and assume that Ψ 1 reaches a negative minimum.

From the jump condition at r * , if this maximum is reached at an interior point, it cannot be at r = r * and the contradiction follows from the Equation. We exclude the case of a negative minimum at R through the same reasons as for p k . It follows that ψ k ψ 1 so that

ω k -ω 1 = ψ 1 (r * ) -ψ k (r * ) 0.
To conclude the proof of Lemma 7, it remains to prove that

ω 1 > 0. ( 63
)
Proof of (63). We define Ψ := u * + ψ 1 .

We note that

- 1 r (r(u * ) ) = λ * + V * - 1 r 2 u * , [(u * ) ](r * ) = [u ](r * ) = u * (r * ). By Hopf's Lemma, u * (R) < 0 and, since u * is C 2 in B * , u * (0) = 0.
We get the following equation on Ψ:

- 1 r (rΨ ) = λ * + V * - 1 r 2 Ψ , [Ψ ](r * ) = 0 , Ψ(0) = 0 , Ψ(R) < 0. Defining Θ := Ψ u * we get - 1 r (rΘ ) = - 1 r 2 Θ + 2 u * u * Θ
and Θ can thus not reach a positive maximum at an interior point. Since it is negative at r = R we get Θ 0 in [0; R]. Furthermore, it is not identically zero since Θ(R) = 0, and the strong maximum principle implies Θ < 0 in (0; R). This gives

Θ(r * ) < 0 or, equivalently ω 1 = -u * (r * )Θ(r * ) > 0
and this concludes the proof of Lemma 7.

Conclusion of the proof

To prove Proposition 1, we simply write

L τ (B * )[g, g] = ∞ k=1 ω k α 2 k + β 2 k by (58)
C ∞ k=1 α 2 k + β 2 k by Lemma 7 = C||g|| 2 L 2 = C|| Φ, ν || 2 L 2 .
The proof of the Proposition is now complete.

Taylor-Lagrange formula and control of the remainder

We now state the main estimate which will enable us to apply the Taylor-Lagrange formula.

Proposition 2. Let M > 0 and η > 0. There exists s ∈ (0; 1) , ε = ε(η) > 0 such that for any

Φ ∈ X 1 (B * ) satisfying ||Φ|| C 1,1 M , ||Φ|| C 1,s ε there holds |L τ (B Φ )[Φ, Φ] -L τ (B * )[Φ, Φ]| η|| Φ, ν || 2 L 2 . (64)
As mentioned earlier, this will prove a quantitative inequality for sets of bounded curvature that are in a C 1,s neighbourhood of B * . Note that working in the C 1,s norm rather than in a C 2,s norm will be enough, since elliptic regularity estimates will prove sufficient for our proofs. The proof of this proposition is technical but not unexpected in this context. We postpone the proof to Appendix C.

Conclusion of the proof of Step 4

Recall that we have defined f (t) := λ(B tΦ ).

Then:

λ(B Φ ) -λ(B * ) = f (1) -f (0) = f (0) (= 0 because B * is critical ) + 1 0 (1 -t)f (t)dt = 1 2 f (0) + 1 0 (1 -t) (f (t) -f (0)) dt C|| Φ, ν || 2 L 2 -η|| Φ, ν || 2 L 2 C 2 || Φ, ν || 2 L 2 C || Φ, ν || 2 L 1 by the Cauchy-Schwarz Inequality = C δ 2 because |B Φ ∆B * | = δ.
whenever ||∇Φ|| L ∞ is small enough. This concludes the proof of Step 4.

A remark on the coercivity norm

The L 2 coercivity established in Proposition 1 is not only sufficient, but also optimal. Indeed, since ψ k is non-negative in (0; R), we immediately have the bound

0 ω k - ∂u * ∂ν .
In other words, the coercivity norm for the second derivative is the L 2 (rather than the H

2 ) norm of the perturbation. This is due to the fact that here, in the context of parametric optimization, it is the value of the shape derivative u rather than the value of its normal derivative that is involved in the second order shape derivative. We not that this is in sharp contrast with classical shape optimization, where the optimization is carried out with respect to the whole domain Ω, and where the coercivity norm is the H 1 2 norm, see [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF] and the references therein.

Step 5: Conclusion of the proof of Theorem 1

We now conclude the proof of Theorem 1. We will argue by contradiction, but let us first fix some notations. For any δ > 0, let V δ be a solution of the variational problem [START_REF] Cianchi | A quantitative polya?szego principle[END_REF]. In the same way we derived the optimality conditions for the radial version of the optimization problem, that is, for the variational problem [START_REF] Caubet | Optimal location of resources for biased movement of species: the 1D case[END_REF], it is easy to see that V δ is equal to 0 or 1 almost everywhere and that, furthermore, if u δ is the associated eigenfunction, that there exists two real numbers µ δ and η δ such that

{V δ = 1} = u δ µ δ ∩ B * ∪ {u δ η δ } ∩ (B * ) c .
We refer to Figure 4 below.

Remark 3. We actually expect that

V δ = H δ
where H δ was defined in Step 3, at least for δ > 0 small enough, in which case Step 4 would prove irrelevant. Put otherwise, we expect the solution to [START_REF] Cianchi | A quantitative polya?szego principle[END_REF] to be a radially symmetric set, given the symmetries properties involved. We were not able to prove this result but we give several numerical simulations in the one-dimensional case, i.e with Ω = (-1; 1), that back this conjecture up. We plot, for several values of δ > 0 and V 0 = 0.6, both V δ and V * -V δ :

- We introduce one last parameter: let ζ δ be the unique real number such that

{u δ > ζ δ } = V 0 . (65)
In the two figures below, we represent the two most extreme cases we might face (note that we always represent sets that are symmetric with respect to the x-axis; this is allowed by Steiner's rearrangement but this property will not be used in what follows) {V δ = 1}

{u δ = ζ δ } {u δ = µ δ } {u δ = η δ } {V δ = 1}
Figure 5: Here, the set V δ is disconnected, and we might compare it with a radial distribution.

To formalize this, we introduce the quantity

f (δ) := |{u δ ζ δ } ∆ ({u δ η δ } ∩ (B * ) c )| . Since |{u δ η δ } ∩ (B * ) c | = δ 2 because V δ ∈ M δ , we have f (δ) δ 2 .
Let us now turn back to the proof of Theorem 1.

To prove Theorem 1, we will as mentioned argue by contradiction: assume that the estimate ( 19) is not valid, that is, there exists a sequence {δ k } k∈IN such that lim k→∞ δ k = 0 and, for any k ∈ IN, δ k > 0 and, furthermore,

lim k→∞ λ(V δ k ) -λ(V * ) δ 2 k = 0.
Since we have, for every k ∈ IN,

f (δ k )
δ k 2 we can also assume that, up to an extraction:

the sequence f (δ k ) δ k k∈IN is converging.
We now establish a dichotomy depending on the limit of f (δ k )

δ k k∈IN
, and distinguish two cases:

1. First case: comparison with a radial distribution

The first case is defined by

f (δ k ) δ k → k→∞ > 0.
In that case, f (δ k ) ∼ k→∞ δ k .We now apply the bathub principle: let E 1 δ k be the solution of

inf E⊂{u δ k >ζ δ k } ,|E|=V0-f (δ k ) - {u δ k ζ δ k }∩E u 2 δ k . If ζ δ k ,1 is defined through {u δ k > ζ δ k ,1 } = V 0 -f (δ k ) then ζ δ k ,1 > ζ δ k and consequently E 1 δ k = {u δ k > ζ δ k ,1 }.
In the same way, we define E 2 δ k as the solution of inf

E⊂ {uδ k >ζ δ k } c ,|E|=f (δ k ) - {u δ k ζ δ k } c ∩E u 2 δ k
and, if we define ζ δ k ,2 through the equation

{ζ δ k > u δ k > ζ δ k ,2 } = f (δ k ) then E 2 δ k = {ζ δ k > u δ k > ζ δ k ,2 } . We replace V δ k by W δ k := χ E 1 δ k + χ E 2 δ k
.

From the bathutb principle,

- Ω V δ k u 2 δ k - Ω W δ k u 2 δ k .
However, W δ k might not satisfy

Ω |W δ k -χ B * | = δ k .
We represent E i δ k , i = 1, 2, below:

Figure 6: An illustration of the process Finally, following the notations of step 3, we recall that A f (δ k ) is defined as

A f (δ k ) := |x| r * -r f (δ k ) ∪ {r * |x| r * + r f (δ k ) } , χ A f (δ k ) ∈ M f (δ k ) .
Our competitor is A f (δ k ) :

Let u * δ k be the Schwarz rearrangement of u δ k . By equimeasurability of the Schwarz rearrangement, we have

A f (δ k ) (u * δ k ) 2 = Ω W δ k u 2 δ k .
By the Polya-Szego Inequality (see [START_REF] Kawohl | Rearrangements and Convexity of Level Sets in PDE[END_REF]),

Ω |∇u * δ k | 2 Ω |∇u δ k | 2 .
Finally, we have established the chain of inequalities

λ(V δ k ) Ω |∇u δ k | 2 - Ω W δ k u 2 δ k Ω |∇u * δ k | 2 - Ω χ A f (δ k ) (u * δ k ) 2 λ A (f (δ k )
by the Rayleigh quotient formulation (3). Now, by [START_REF] Dambrine | On variations of the shape hessian and sufficient conditions for the stability of critical shapes[END_REF],

λ A (f (δ k ) λ * + Cf (δ k ) 2
and thus, since

f (δ k ) ∼ k→∞ δ k we have λ(V δ k ) -λ * C 2 δ k 2
which gives the required contradiction and concludes the proof.

Second case: comparison with a normal deformation

The second case is defined by

f (δ k ) δ k → k→∞ 0. ( 66
)
In this case, we use Step 4 of the proof, i.e the quantitative inequality for normal deformations of the ball.

Let us replace V δ k with W δ k := χ {uδ k >ζ δ k } .

Recall that ζ δ k was defined in such a way that W δ k ∈ M(Ω). By the bathtub principle,

λ(V δ k ) λ(W δ k ).
Furthermore, Condition (66) implies

|{u δ k > ζ δ k } ∆B * | = δ k + o k→∞ (δ k ). Indeed, δ k 2 = |{u δ k > η δ k } ∩ (B * ) c | = |{u δ k > ζ δ k } ∩ (B * ) c | + f (δ k ).
Finally, standard elliptic estimates imply (in dimension 2 and 3) that We start by noticing that (67) implies

u δ k C 1,
d H ({u δ k > ζ δ k } , B * ) → k→∞ 0, (68)
where d H is the Hausdorff distance. We then argue by contradiction and assume that, a subsequence of {δ k } k∈IN there exists x δ k ∈ ∂B * , t 1 = t 2 ∈ IR such that

x δ k + t j ν(x δ k ) ∈ ∂ {u δ k > ζ δ k } , j = 1, 2.
It follows that

u δ (x δ k + t 1 ν(x δ k )) = u δ k (x δ k + t 2 ν(x δ k ))
and by the intermediate value Theorem and (68), there exists t δ k ∈ IR such that

∇u δ (x δ k + t δ k ν(x δ k )) , ν(x δ k ) = 0 , t δ k → k→∞ 0.
By passing to the limit in this equation up to a subsequence, there exists a point x * ∈ ∂B * such that ∇u * (x * ) , ν(x * ) = 0. This is a contradiction since ∂u * ∂ν ∂B * = 0.

We can then say that ∂ {u δ k > ζ δ k } is the graph of a function ϕ δ k over ∂B * . Besides, the Convergence result (67) implies that

ϕ δ k C 1 (∂B * ) → k→∞ 0.
Finally, since the set {u δ k µ δ k } converges in the C 1,s topology to B * , there exists a uniform radius r > 0 such that, for any x ∈ ∂{u δ k µ δ k }, there exists y x satisfying

x ∈ B(y x , r) , B(y x , r) ⊂ {u δ k µ δ k }.

Since V δ k is constant in B(y x , r), we can apply elliptic regularity results to get a uniform

C 2 norm on u δ k in {u δ k µ δ k }: there exists M > 0 such that, for any x ∈ {u δ k µ δ k }, dist(x , ∂{u δ k , µ δ k }) r, |∇ 2 u(x)| M . Hence, the curvature of {u δ k µ δ k } is uniformly bounded by some constant M .
To prove that the curvature of {u δ k > ζ δ k } is uniformly bounded as well, we note the following fact: for any

x ∈ ∂{u δ k > ζ δ k }, let x δ k be its orthogonal projection on {u δ k µ δ k }. Then, µ δ k -ζ δ k ∼ k→∞ |x δ k -x| ∂u δ k ∂ν (x δ k )
and so the map

∂{u δ k ζ δ k } (x, y) → |x-x δ k | |y-y δ k | converges
λ(V δ k ) > λ(W δ k ) λ * + C |{u δ k > ζ δ k } ∆B * | 2 by Step 4 λ * + C(δ k + o k→∞ (δ k )) 2 λ * + C δ k 2 .
This gives a contradiction.

The proof of Theorem 1 is now complete.

3 Concluding remarks and conjecture

Extension to other domains

We do believe that this quantitative inequality is valid not only in the ball but for more general domains. Let, for any domain Ω, V Ω be a solution of [START_REF] Belgacem | Elliptic Boundary Value Problems with Indefinite Weights, Variational Formulations of the Principal Eigenvalue, and Applications[END_REF]. Let u Ω be the associated eigenfunction. By the bathtub principle, it is easy to see that there exists µ Ω ∈ IR such that

V Ω = χ {uΩ µΩ} = χ EΩ .
We give the following conjecture:

Conjecture 1. Assume that 1. The minimizer is regular in the sense that ∂uΩ ∂ν -C < 0 on ∂E Ω ,

2. E Ω is a non-degenerate shape minimizer: for any admissible variation Φ ∈ X 1 (E Ω ), if L τ is the associated lagrangian, there holds

L τ (E Ω )[Φ, Φ] > 0.
Then there exists a parameter η > 0 such that, for any

V ∈ M(Ω), ||V -V Ω || L 1 (Ω) η ⇒ λ(V ) -λ(V * ) C||V -V * || 2 L 1 (Ω) .
Here the main difficulty lies not only in the quantitative inequality for normal perturbations of the domain (Step 4 of the proof of Theorem 1) but also in the quantitative inequality for possibly disconnected competitors (Step 3 of the proof). Indeed, since the parametric derivatives uΩ are no longer constant on the boundary of the set E Ω , the approach used in Step 3 might fail. We note, however, that the infinitesimal quantitative bound

λ(V Ω )[h] C||h|| 2 L 1 (Ω)
still holds. To see why, we notice that

λ(V Ω )[h] = - Ω hu 2 Ω
and consider, for a parameter δ, the solution h δ of min

h admissible at VΩ ,||h|| L 1 (Ω) =δ λ(V Ω )[h].
By the bathtub principle, h δ can be written for any δ as a level set of u δ and, for δ small enough, one can prove that h δ writes as follows: for some constant C that depends on inf ∂uΩ ∂ν . Thus, the infinitesimal inequality seems valid. However, it seems complicated to go further using only this information, since the parametric derivatives u are no longer constant on ∂E Ω .

h δ = χ E + δ -χ E - δ , where E + δ ⊂ E c Ω and E - δ ⊂ E Ω and

Other constraints

It would be relevant to consider perimeter constraints instead of volume constraints, but we expect the behaviour of the sequences of solutions to the auxiliary problems to be quite different. We nonetheless believe that the free boundary techniques used in [START_REF] Brasco | Faber-krahn inequalities in sharp quantitative form[END_REF] might apply directly to get regularity.

For more on these notions, we refer to [START_REF] Henrot | Shape Variation and Optimization[END_REF]. Obivously v t ∈ W 1,2 0 (Ω). The weak formulation on u t writes: for any ϕ ∈ W 1,2 0 (Ω),

B ∇u t , ∇ϕ = λ t B u t ϕ + Et u t ϕ.
We do the change of variables x = T t (y), so that, for any test function ϕ,

B A t ∇v t , ∇ϕ = λ t B v t ϕJ t + E v t ϕJ t . (69)
It is known that

J (x) := ∂J t ∂t t=0 (t, x) = ∇ • Φ,
and that

A(x) := ∂A t ∂t t=0 (t, x) = (∇ • Φ)I n -∇Φ + (∇Φ) T .
We recall that A has the following property: if Φ 1 and Φ 2 are two vector fields, there holds

AΦ 1 , Φ 2 = ∇ • ( Φ 1 , Φ 2 Φ) -∇(Φ • Φ 1 ), Φ 2 -∇(Φ • Φ 2 ), Φ 1 . (70)
We differentiate Equation (69) with respect to t to get the following equation on u:

B ∇ϕ, ∇ u + A∇u 0 = λ B u 0 ϕ + λ 0 B J u 0 ϕ + λ 0 B uϕ + E uϕ + E J (x)u 0 ϕ. (71)
Through Property (70) we get A∇u 0 , ∇ϕ = ∇ • ( ∇u 0 , ∇ϕ Φ) -∇( Φ, ∇u 0 ), ∇ϕ -∇( Φ, ∇ϕ ), ∇u 0 .

We deal with these three terms separately: from the divergence Formula

B ∇ • ( ∇u 0 , ∇ϕ Φ) = - ∂E [ ∇u 0 , ∇ϕ ] Φ, ν .
We do not touch the second term. The third term is dealt with using the weak equation on u 0 :

B ∇( Φ, ∇ϕ ), ∇u 0 = λ 0 B Φ, ∇ϕ u 0 + E u 0 Φ, ∇ϕ - ∂E [ ∇u 0 , ∇ϕ ] Φ, ν . Hence B A∇u 0 , ∇ϕ = B ∇ • ( ∇u 0 , ∇ϕ Φ) -∇( Φ, ∇u 0 ), ∇ϕ -∇( Φ, ∇ϕ ), ∇u 0 = - B ∇( Φ, ∇u 0 ), ∇ϕ -λ 0 B Φ, ∇ϕ u 0 - E u 0 Φ, ∇ϕ . 40
The left hand term of (71) becomes

B ∇ u + A∇u 0 , ∇ϕ = B ∇ u -Φ • ∇u 0 , ∇ϕ -λ 0 B Φ, ∇ϕ u 0 - E u 0 Φ, ϕ . Thus B ∇ u -Φ • ∇u 0 , ∇ϕ -λ 0 B Φ, ∇ϕ u 0 - E u 0 Φ, ∇ϕ = λ B u 0 ϕ + λ 0 B J (x)u 0 ϕ + λ 0 B uϕ + E uϕ + E J (x)u 0 ϕ.
By rearranging the terms, we get

B ∇ u -Φ • ∇u 0 , ∇ϕ = + λ B u 0 ϕ + λ 0 B J (x)u 0 ϕ + B Φ, ∇ϕ u 0 +λ 0 B uϕ + E uϕ + E J (x)u 0 ϕ + E u 0 Φ, ∇ϕ .
However, since J (x) = ∇ • Φ(x), we have We turn back to the shape derivative; recall that it is defined as u := u -Φ, ∇u . Obtaining the jump condition on u is done in the same way as in [START_REF] Dambrine | On the shape sensitivity of the first dirichlet eigenvalue for two-phase problems[END_REF].

J (x)ϕ + Φ, ∇ϕ(x) = ∇ • (ϕΦ) .

The previous equation rewrites

B.3 Gâteaux-differentiability of the eigenvalue

The parametric differentiability is also proved using the implicit function theorem applied to the following map:

G : L ∞ (Ω) × W 1,2 0 (Ω) × IR → W -1,2 (Ω) × IR, (h, v, λ) → -∆v -λv -(V + h)v, Ω v 2 -1 .
The invertibility of the differential follows from the same arguments as the ones used to prove the invertibility of DF in the previous section.

C Proof of Proposition 2

Proof of Proposition 2. We can not apply in a straightforward manner the methods of [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF], which are well-suited for the proof of a convergence in the H It is known (see [START_REF] Dambrine | Stability in shape optimization with second variation[END_REF]Lemma 4.8]) that

J Ω/Σ (Φ) -1 L ∞ C||Φ|| W 1,∞ , ||A Φ -1|| L ∞ C||Φ|| W 1,∞ . (72)
We define u 0 as the eigenfunction asociated with B * and u 0 the shape derivative of u 0 in the direction Φ.

Finally let u Φ be the shape derivative in the direction Φ and û Φ := u Φ • T Φ . Let H Φ be the mean curvature of B Φ . Using the change of variable y = T Φ (x), the fact that Φ is normal to B * and the value of the Lagrange multiplier τ given by (47), we get

L τ (B Φ )[Φ, Φ] = -2 ∂B * J Σ (Φ)û Φ û Φ Φ, ν + ∂B * J Σ (Φ) -Ĥ û2 Φ -2û Φ ∂ ûΦ ∂ν Φ, ν 2 -τ ∂B * Ĥ Φ, ν 2 = -2 ∂B * J Σ (Φ)û Φ û Φ Φ, ν + ∂B * J Σ (Φ) Ĥ(û 2 Φ -u 2 0 ) -2û Φ ∂ ûΦ ∂ν Φ, ν 2 .
Hence we have

L τ (B Φ )[Φ, Φ] -L τ (B * )[Φ, Φ] = -2 ∂B * (J Σ (Φ)û Φ û Φ -u 0 u 0 ) Φ, ν + ∂B * Ĥ(J Σ (Φ)û 2 Φ -u 2 0 ) Φ, ν 2 + ∂B * 2u 0 ∂u 0 ∂ν -2J Σ (Φ)û Φ ∂ ûΦ ∂ν Φ, ν 2 .
(73)

We will prove the Proposition using the following estimates Claim 4. For any η > 0 there exists ε > 0 such that, for any Φ satisfying

||Φ|| C 1 ε there holds 1. ||û Φ -u * || C 1 (Ω) η, (74) 2
. ||û Φ -u 0 || W 1,2 0 η || Φ, ν || L 2 (Σ) . ( 75
)
Proof of Claim 4. Estimate (74) follows from a simple contradiction argument and by using the fact that, if a sequence {Φ k } k∈IN converges in the C 1 norm to 0, then u Φ k converges, in every C 1,s (Ω) (s < 1) to u 0 . To prove (75), we first prove that there exists a constant M such that

||û Φ || W 1,2 0 M || Φ, ν || L 2 (Σ) . (76)
By the change of variable y := T Φ (x), we see that û Φ satisfies

-∇ • A Φ ∇û Φ = J Ω (Φ) (λ Φ û Φ + (V * )û Φ + λ Φ ûΦ ) , [A Φ ∂ ν û Φ ] = -J Σ Φ, ν ûΦ (77)
with homogeneous Dirichlet boundary conditions. The orthogonality conditions gives Ω J Ω (Φ)û Φ û Φ = 0 and we will use a Spectral Gap Estimate (82) combined with a bootstrap argument.

Spectral gap estimate For any V ∈ M(B), λ(V ) was defined as the first eigenvalue of the operator L V defined in [START_REF] Acerbi | Minimality via second variation for a nonlocal isoperimetric problem[END_REF]. We recalled in the Introduction that this eigenvalue is simple. Let, for any V ∈ M(Ω), λ 2 (V ) > λ(V ) and u 2,V be the second eigenvalue and an associated eigenfunction (we choose a L 2 normalization). We claim there exists ω > 0 such that, for any V ∈ M(Ω),

ω λ 2 (V ) -λ(V ). ( 78
)
To prove this, we use a direct argument. Let S(V ) := λ 2 (V ) -λ(V ) be the spectral gap associated with V . We consider a minimizing sequence {V k } k∈IN ∈ M(Ω) (the radiality assumption is not necessary here) which, up to a subsequence, converges weakly in L ∞ - * to some V ∞ ∈ M(Ω). It is standard to see that

λ(V k ) → k→∞ λ(V ∞ ) , u V k → k→∞ u
V∞ strongly in L 2 (B), weakly in W 1,2 0 (B).

By the Spectral gap estimate, using the fact that eigenfunctions are uniformly bounded and by continuity of the trace operator we get the existence of a constant M such that

Ω (û Φ ) 2 M || Φ, ν || L 2 (∂B * ) ||û Φ || W 1,2 0 (Ω) . (83)
We rewrite ||û Φ || W 1,2 0 (Ω) = ||û Φ || L 2 (Ω) + ||∇û Φ || L 2 (Ω) . By the shape differentiability of E → (λ(E), u E ), there exists C such that

||∇û Φ || L 2 (Ω) C
for any Φ such that ||Φ|| W 1,∞ 1. We then let X := ||û Φ || L 2 (Ω) . The estimate (83) rewrites

X 2 M || Φ, ν || L 2 (∂Ω) X + M C|| Φ, ν || L 2 (∂Ω) ,
from where it follows that there exists M > 0 such that

||û Φ || L 2 (Ω) M || Φ, ν || L 2 (∂Ω) .
We now multiply (76) by û Φ and integrate by part. Using the continuitiy of the trace operator, this gives, for some constant M ,

Ω |∇û Φ | 2 C|| Φ, ν || L 2 (∂Ω) + || Φ, ν || L 2 (∂Ω) ||∇û Φ || L 2 (Ω) + || Φ, ν || L 2 (∂Ω)

1 Introduction 1 . 1 1 Introduction 1 . 1

 111111 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Mathematical setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Quantitative inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 A comment on parametric and shape derivatives . . . . . . . . . . . . . . . 1.3.4 A remark on the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Bibliographical references . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Quantitative spectral inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Mathematical biology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structure of the paper

Figure 3 :

 3 Figure 3: A perturbation of the ball which can not be seen as the graph of a function

Figure 4 :

 4 Figure 4: Here, the set V δ is connected, and we might compare it with a normal deformation of B * .

  ∇u 0 , because u 0 satisfies homogeneous Dirichlet boundary conditions. In the same wayE J (x)u 0 ϕ + E Φ, ∇ϕ u 0 = B ∇ • (Φϕ)u 0 = -E ϕ Φ, ∇u 0 + ∂E Φ, ν u 0 ϕ.

  Thus there appears that u solves-∆u = λ u 0 + λ 0 u 1 + V ualong with Dirichlet boundary conditions and ∂u ∂r = -Φ, ν u 0 .

1 2

 1 topology. Some minor adjustments are in order. Let us define T Φ := (Id + Φ) and, for any function f : Ω → IR,f := f • T Φ .We define the surface JacobianJ Σ (Φ) := det(∇T Φ ) t ∇T -1 Φ ν ,the volume JacobianJ Ω (Φ) := det (∇Φ) and, finally A Φ := J Ω (Φ)(Id + ∇Φ) -1 (Id + t ∇Φ) -1 .

3 2 which|| 2 L 2 2 .

 32222 in turn yields, using the same arguments,||∇û Φ || L 2 (Ω) || Φ, ν || L 2 (∂Ω) .We use this in (83), giving||û Φ (Ω) M || Φ, ν || L 2 (∂Ω) ||û Φ || L 2 (Ω) + || Φ, ν || L 2 (∂Ω) 3 This yields ||û Φ || L 2 (Ω) M || Φ, ν || L 2 (∂Ω)and, finally, from the weak formulation of the equation,||∇û Φ || L 2 (Ω) M || Φ, ν || L 2 (∂Ω) .Proof of (75) We now turn to the proof of the continuity estimate (75), for which we will apply the same kind of bootstrap arguments, combined with a version of the splitting method, see[START_REF] Dambrine | Stability in shape optimization with second variation[END_REF] Lemma 4.10].Let us define H Φ as the solution of   -∆H Φ = V Φ H Φ , ∂HΦ ∂ν= u Φ Φ, ν Φ on ∂E t , H Φ = 0 on ∂Ω.

  that λ Φ = -∂Et u 2 Φ Φ, ν Φ = λ Φ Ω H Φ u Φ .

  S * ∓ 2u δ,t |∇u δ,t || S

	hu 2 δ,t
	B
	= -u 2 δ,t S *	B	h δ = 0 because	B	h = 0
	+
	I -

* dist(x; S * ) + o (dist(x; S * )) , and o (dist(x; S * )) is uniform in δ > 0 small enough and t ∈ [0; 1] by Estimate (39). This Taylor expansion gives λδ,t =δ 2u δ,t |∇u δ,t || S * dist(x; S * ) + o I - δ dist(x; S * ) + I + δ 2u δ,t |∇u δ,t || S * dist(x; S * ) + o I + δ dist(x; S * ) where the o I + δ dist(x; S * ) are uniform in t ∈ [0; 1] and δ. Furthermore, u δ,t |∇u δ,t | S * C > 0 for some constant C > 0 independent of δ and t by Estimate (41). Hence λδ,t ∼ δ→0 2u δ,t |∇u δ,t || S * I + δ dist(x; S * ) +

  ∂ν | ∂B * = 0 and ζ δ k → | ∂B * it follows that ∂ {u δ k > ζ δ k } is a C 1 hypersurface by the implicit function Theorem. It remains to prove that ∂ {u δ k > ζ δ k } is a graph above ∂B * .

	and, since ∂u
	s (Ω) → k→∞	u *	(67)

* k→∞ u *

  uniformly to 1. {u δ k > ζ δ k } can thus be described, asymptotically, as {u δ k µ δ k } + B(0; t δ k ), so that it also has a uniformly bounded curvature. Remark 4. We could have worked directly with {u δ k µ δ k } + B(0; t δ k ), by choosing a suitable t δ k but in this context, it seemed more relevant to work with level sets.

	We can hence apply Step 4:

  can be described as follows: if ν is the unit normal vector to ∂E. It remains to apply the methods of Step 3 of the Proof of Theorem 1 and to do a Taylor expansion of -u2 Ω at ∂E Ω to get λ(V Ω )[h δ ] Cδ 2

	E Ω ,
	E ± δ := {x ± tν(x) , t ∈ (0; t δ ± (x))}.
	We can then prove that
	t δ ± δ	→ δ→0	f ± > 0
	uniformly in x ∈

Acknowledgment. I. Mazari was partially supported by the Project "Analysis and simulation of optimal shapes -application to lifesciences" of the Paris City Hall and by the ANR Project "Optimisation de forme -SHAPO".

The author would like to thank D. Bucur and J. Lamboley for the useful discussions they had with him, and express his gratitude to D. Ruiz-Balet for his help in obtaining the numerical simulations of Remark 3.

The author would also like to warmly thank the anonymous referee for his or her insightful comments and advice.

A Proof of Lemma 1

We briefly recall that the Schwarz rearrangement of a function u ∈ W 1,2 0 (B), u 0 is defined as the only radially symmetric non-increasing function u * such that, for any t ∈ IR, |{u t}| = |{u * t}| .

Proof of Lemma 1. We use the Polya-Szego Inequality for the Schwarz rearrangement: for any u ∈ W 1,2 0 (B) , u 0,

We also use the Hardy-Littlewood Inequality: for any u, v ∈ L 2 (Ω),

and the equimeasurability of the rearrangement:

We refer to [START_REF] Kawohl | Rearrangements and Convexity of Level Sets in PDE[END_REF] for proofs. Using the Rayleigh quotient formulation (3), for any V ∈ M(B),

This also proves that u V * = u * V * . Since the eigenvalue is simple, the eigenfunction is radially symmetric. The fact that it is decreasing follows from the Equation satisfied by u V * in polar coordinates.

B Proof of the shape differentiability of λ B.1 Proof of the shape differentiability Proof of the shape differentiability. Let E be a regular subdomain of B, (u 0 , λ 0 ) be the eigenpair associated with V := χ E , and let Φ be an admissible vector field at E. Let T Φ := (Id + Φ) and E * Φ := T Φ (E). Let u Φ be the eigenvalue associated with V Φ := χ EΦ and λ Φ be the associated eigenvalue. If we introduce

then the weak formulation of the equation on u Φ is: for any v ∈ W 1,2 0 (Ω),

We define the map F in the following way:

It is clear from the definition of the eigenvalue that

Furthermore, the same arguments as in [START_REF] Dambrine | On the shape sensitivity of the first dirichlet eigenvalue for two-phase problems[END_REF]Lemma 2.3]

To prove that this differential is invertible, it suffices to show that, if (z, k) ∈ W -1,2 (Ω) × IR, then there exists a unique couple (w, µ) such that

By the Fredholm alternative, we know that we must have

There exists a solution w 1 to the equation

We fix such a solution. Any other solution is of the form w = w 1 + tu 0 for a real parameter t. We look for such a t. From the equation 2

hence the couple (w, µ) is uniquely determined. From the implicit function theorem, the map

B.2 Computation of the first order shape derivative

Proof of Lemma 6. Let Φ be a smooth vector field at E and E t = T t (E) where T t := Id + tΦ. We define J t := J Ω (tΦ) and A t := A tΦ . The other notations are the same as in the previous paragraph. Let (λ t , u t ) be the eigencouple associated with V t := χ Et . We first define

The derivative of v t with respect to t will be denoted u. This is the material derivative, while we aim at computing the shape derivative u defined as

The only part which is not completely classical is to prove that

However, for any k ∈ IN, λ 2 (V k ) is defined as

where u ⊥ is the subspace of functions that are L 2 -orthogonal to u, and u 2,V k is defined as a minimizer for this problem (there a possibly multiple eigenfunctions). In the same way we proved that λ(V ) is uniformly bounded in V , one proves that λ 2 (V ) is uniformly bounded inV . Let λ 2,∞ be such that

Standard elliptic estimates prove that there exists a function u 2,∞ ∈ W 1,2 0 (Ω) such that u 2,V k → k→∞ u 2,∞ strongly in L 2 (B) and weakly in W 1,2 0 (B). Passing to the limit in

Passing to the limit in the weak formulation of the equation on u 2,V k proves that u 2,∞ is an eigenfunction of L V∞ associated with λ 2,∞ . It follows from the orthogonality relation (81) that

As a consequence of the spectral gap estimate (78), we get the following estimate:

Indeed, let V ∈ M(Ω) and u ∈ u V ⊥ , u = 0. Then, by the Rayleigh quotient formulation on λ 2 (V ), see Equation (80),

which is exactly the desired conclusion.

Proof of (76) First of all, multiplying (77) by û Φ and integrating by parts gives

We can prove using the same bootstrap arguments used to prove (76) that

Indeed, multiplying (85) by H Φ , doing a change of variables and integrating by parts gives

and, by the variational formulation of the eigenvalue,

We then use the same bootstrap argument: we first prove that this implies

and plug this estimate in the weak formulation of the equation. The conclusion follows.

We turn back to (75). Let π Φ be the orthogonal projection on u Φ ⊥ . We decompose u Φ as

Thanks to the Fredholm alternative, such a ξ Φ exists and is uniquely defined. We now prove that ĤΦ -H 0

for ||Φ|| C 1 small enough. To that end, we define

Direct computation shows that

along with Dirichlet boundary conditions and

We proceed in the same fashion: we first multiply the equation on H Φ by H Φ , integrate by parts and use the variational formulation of the eigenvalue to get

∂Ω) up to a multiplicative constant. This first gives, using (74),

We then apply the same bootstrap method to get the desired conclusion. Finally, we need to show the following estimate, which will conclude the proof:

for ||Φ|| C 1 small enough. However, this follows from the same arguments as in [22, Lemma 4.10, Paragraph 3 of the proof] and from the bootstrap strategy already used.

Finally, going back to (73), it suffices to use the continuity of the trace to control the terms involving u Φ and Estimates (74)-(75) to conclude the proof of Proposition 2.