Jérémy Dalphin
email: jdalphin@dim.uchile.cl

Shape derivatives of the probability to nd a xed number of electrons chemically characterized by a wave function

Keywords: shape optimization, shape derivatives, volume integral, maximal probability domains, geometry of wave functions, quantum chemistry AMS classication : 49K40, 49M15, 49M05, 81Q99, 92E99, 81V99, 51M04, 51M16

In Quantum Chemistry, researchers are interested in nding new ways to describe well the electronic structures of molecules and their interactions. The model of Maximal Probability Domains (MPDs) is a developing method based on probabilities that allows such a geometrical and spatial characterization of the electronic structures of chemical systems.

In this article, we consider quantum systems of n electrons chemically characterized by general wave functions. For any integer k 1, we derive a formula for the k-th-order shape derivative of the functional pν : Ω → pν (Ω), with pν (Ω) the probability to nd exactly a xed number ν of electrons in a given spatial region Ω ⊆ R 3 , where exactly means that the n -ν remaining ones are located in the complement R 3 \Ω.

This explicit formula is computable by Quantum Monte-Carlo methods and it holds true with respect to the W 1,∞ -perturbations of a measurable domain for H k -regular wave functions. Then, by restricting our analysis to the rst-and second-order shape derivatives, we can make our statement more precise with respect to the regularity of the domain, and recover the usual structure expected from shape derivatives.

The main ingredient of the proof consists in generalizing at any higher order the well-known expressions for the rst-and second-order shape derivatives of a volume integral. Although we only need to assume that the domain is measurable to get the shape dierentiability of a volume integral at any order, we also prove that the C 1,1 -regularity is enough to provide a notion of partial derivative with respect to the domain at any order (shape gradient, Hessian,...).

Introduction

On the one hand, the traditional chemical intuition i.e. the way chemists understand how molecules interact together has been deeply inuenced by a localized vision of electrons around the cores. Indeed, it yields to fruitful concepts [START_REF] Lewis | The atom and the molecule[END_REF][START_REF] Pauling | The nature of the chemical bond: application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules[END_REF][START_REF] Shaik | The charge-shift bonding concept: electron-pair bonds with very large ionic-covalent resonance energies[END_REF] rmly rooted to the models because it can simply explain many dierent experimental manifestations. On the other hand, Quantum Mechanics [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF][START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF][START_REF] Szabo | Modern Quantum Chemistry: introduction to advanced electronic structure theory[END_REF] allows the electrons to be delocalized over the whole space. Indeed, a chemical system of n electrons is completely characterized by its wave function, a priori dened everywhere.

Hence, there is a loss of chemical informations that Quantum Chemistry tries to recover in several manners. Interpretative methods (valence bond theory, molecular orbitals) work in the Fock space when the correlation between electrons is small, while topological approaches try to partition directly the physical space into regions with a chemical meaning [2,3,[START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF][START_REF] Politzer | Separation of core and valence regions in atoms[END_REF][START_REF] Savin | On the signicance of ELF basins[END_REF][START_REF] Silvi | Classication of chemical bonds based on topological analysis of electron localization functions[END_REF].

One way to reconnect the usual expectations of chemists with the results of accurate quantum mechanical calculations consists in removing the problematical high-dimensionality of the wave function by averaging it correctly over the positions of electrons [START_REF] Chamorro | Electron probability distribution in AIM and ELF basins[END_REF][START_REF] Francisco | Electron number probability distributions for correlated wave functions[END_REF][START_REF] Pendás | Pauling resonant structures in real space through electron number probability distributions[END_REF]. More precisely, computing the probability p ν (Ω) to nd exactly a xed number ν of electrons in a given spatial region Ω ⊆ R 3 , where exactly means that the n -ν remaining ones are located in the complement part R 3 \Ω, one can try to solve the following shape optimization problem:

sup Ω⊆R 3 p ν (Ω) . (1)
Suggested by Savin in [START_REF] Savin | Probability distributions and valence shells in atoms[END_REF], the model of Maximal Probability Domains (MPDs) i.e. searching for the local/global maximizers and the critical points of (1) is a developing method based on probabilities that allows a geometrical and spatial characterization of the electronic structures of molecules and their interactions. Indeed, it has shown to provide vivid images of cores and valence regions of atoms [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF][START_REF] Savin | Probability distributions and valence shells in atoms[END_REF], lone and bonding pairs [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF], and domains in which can move the electrons in a simple molecule [START_REF] Gallegos | Maximal probability domains in linear molecules[END_REF][START_REF] Lopes | Understanding maximum probability domains with simple models[END_REF][START_REF] Menéndez | On the stability of some analytically solvable maximum probability domains[END_REF], a liquid [1], a crystal [START_REF] Causà | Maximum probability domains in crystals: the rock-salt structure[END_REF][START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF], or in an inorganic compound [START_REF] Causà | The bond analysis techniques (ELF and maximum probability domains): application to a family of models relevant to bio-inorganic chemistry[END_REF].

Therefore, MPDs may become a rigorous entry point to recover standard chemical concepts from the quantum informations of the systems gathered in the wave functions. For example, the domains that locally maximize the probability to nd exactly two electrons can be directly related to the Lewis' concept of electron pair [START_REF] Lewis | The atom and the molecule[END_REF] and it provides a visual representation of this chemical interaction in the physical three-dimensional space.

The mathematical existence and regularity of maximizers for (1) are dicult and open problems, even for simple analytic wave functions such as a two-electron molecule. Indeed, the direct method from Calculus of Variations does not apply here. Roughly speaking, there is a lack of continuity and compactness due to the poor control we have on the perimeter of a minimizing sequence. The boundary can oscillate severely, reveal some cracks and cusps, or simply become unbounded.

From a numerical point of view, it is still an on-going eort to develop algorithms and programs that are able to eciently optimize the domains solving (1). The gradient and Newton methods heavily rely on the concept of shape derivatives [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF][START_REF] Menéndez | On the stability of some analytically solvable maximum probability domains[END_REF], where mesh adaptation techniques seem necessary to ensure a certain condence in the numerical MPDs obtained. We also mention [START_REF] Scemama | Investigating the volume maximizing the probability of nding ν electrons from Variational Monte Carlo data[END_REF][START_REF] Scemama | Maximum probability domains from Quantum Monte Carlo calculations[END_REF] where a Quantum Monte-Carlo approach is used to obtain some MPDs.

The goal of this article is to properly derive formulas for the shape derivatives of the functional p ν : Ω → p ν (Ω) with general wave functions. To our knowledge, such a theoretical study has not been carried out in its generality, although some expressions were obtained at the Hartree-Fock level [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF][START_REF] Menéndez | On the stability of some analytically solvable maximum probability domains[END_REF]. In particular, the rst-and second-order shape derivatives are fundamental in the numerical implementation but also to gain theoretical informations about the nature of MPDs.

In this paper, our rst main contribution is to study the (Fréchet) dierentiability properties of the map p ν,Ω : θ → p ν [(I + θ)(Ω)] associated with the shape functional p ν : Ω → p ν (Ω) for general wave functions Ψ. Under the H k -regularity of Ψ, we get that p ν,Ω is of class C k around the origin for any integer k 0 and for any measurable subset Ω of R 3 . The results with their precise references in the text are sum up in Table 1, where B 0,1 refers to the set of Lipschitz contractions.

Ω ⊆ R 3 Ψ θ : R 3 → R 3 Regularity of p ν,Ω : θ → p ν [(I + θ) (Ω)] Proof Measurable L 2
The map p ν : Ω → p ν (Ω) is well dened. Denition 2.2 Measurable L 2 C 0,1 p ν,Ω is well dened on B 0,1 . Lemma 2.6 Measurable

L 2 W 1,∞ p ν,Ω is of class C 0 on B 0,1 ∩ W 1,∞ . Lemma 2.6 Measurable H 1 W 1,∞ p ν,Ω is of class C 1 on B 0,1 ∩ W 1,∞ . Corollary 2.9 Lipschitz H 1 W 1,∞
p ν has a well-dened shape gradient. Theorem 2.8 Measurable

H 2 W 1,∞ p ν,Ω is of class C 2 on B 0,1 ∩ W 1,∞ . Corollary 2.11 Lipschitz H 2 W 1,∞ ∩ C 1 p ν,Ω is of class C 2 on B 0,1 ∩ W 1,∞ ∩ C 1 . Corollary 2.11 C 1,1 -domain H 2 W 1,∞ ∩ C 1
p ν has a well-dened shape Hessian. Theorem 2.10 Measurable

H k W 1,∞ p ν,Ω is of class C k on B 0,1 ∩ W 1,∞ . Theorem 2.7
Table 1: Summary of the regularity results concerning the functional p ν,Ω : θ → p ν [(I + θ)(Ω)].

The main achievement of Theorem 2.7 is to get an explicit formula [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF] for the k-th-order shape derivative of p ν i.e. the k-th-order (Fréchet) dierential of p ν,Ω at the origin for any integer k 1. The counterpart of this general result is the poor structure we get for the shape derivatives of p ν . Hence, by restricting our analysis to the low-order ones, our second main contribution consists in recovering the shape gradient and Hessian form that are expected from more regular domains [START_REF] Novruzi | Structure of shape derivatives[END_REF] [22, Section 5.9] [START_REF] Bucur | Anatomy of the shape Hessian via lie brackets[END_REF].

In Theorem 2.8, we show that if the domain has a Lipschitz boundary, then the functional p ν : Ω → p ν (Ω) of Denition 2.2 has a rst-order shape derivative of the following form:

∀θ ∈ W 1,∞ R 3 , R 3 , D 0 p ν,Ω (θ) = ∂Ω ∂p ν ∂Ω (x) θ n (x) dA (x) , (2)
where the integration on the boundary ∂Ω is done with respect to the two-dimensional Hausdor measure referred to as A(•), where (•) n := (•) | n Ω is the normal component of a vector eld, with n Ω (x) the unit vector normal to the boundary ∂Ω at the point x and pointing outwards Ω, and where the following map is well dened by [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF]:

∂p ν ∂Ω : ∂Ω -→ R.
It depends on the domain Ω but not on the perturbation θ. Hence, by analogy with the nitedimensional case, it is called the shape gradient of p ν . Similarly, in Theorem 2.10, we prove that if the domain has a C 1,1 -boundary, then the functional p ν : Ω → p ν (Ω) of Denition 2.2 has a second-order shape derivative of the following form:

∀(θ, θ) ∈ W 1,∞ ∩ C 1 2 , D 2 0 p ν,Ω (θ, θ) = ∂Ω ∂ 2 p ν ∂Ω 2 (x) θ n (x) θn (x) dA (x) + ∂Ω ∂Ω
K Ω (x, y) θ n (x) θn (y) dA (x) dA (y)

- ∂Ω ∂p ν ∂Ω (x) Z[θ, θ] (x) dA (x) . (3)
We mention that in (3) the perturbations θ and θ must be continuously dierentiable since the expression of Z involves the derivatives of θ and θ whose values are computed on the boundary. The rst term of (3) can be interpreted as the Hessian part of the second-order shape derivative. Indeed, the following map is well dened by [START_REF] Savin | On the signicance of ELF basins[END_REF]:

∂ 2 p ν ∂Ω 2 : ∂Ω -→ R.
It depends on Ω but not on the perturbations (θ, θ). By analogy with the nite-dimensional case, it is called the shape Hessian of p ν . However, the second term in (3) also plays an important role. It has the form of a kernel and the following map is well dened by [START_REF] Savin | Probability distributions and valence shells in atoms[END_REF]:

K Ω : ∂Ω × ∂Ω -→ R.
Again, the kernel K Ω depends on Ω but not on the perturbations θ and θ. Finally, note that the last term in (3) depends on the shape gradient. In particular, if Ω is a critical shape for p ν i.e. if D 0 p ν,Ω ≡ 0, then ∂pν ∂Ω = 0 and this term is equal to zero. It also depends on the vector eld: The method used for the proof of Theorem 2.7 consists in expressing p ν,Ω as a volume integral on an higher-dimensional space. The rst-and second-order shape derivatives of a volume integral are well known in the context of shape calculus [START_REF] Delfour | Shapes and geometries: metrics, analysis, dierential calculus, and optimization[END_REF]Chapter 9] [22, Chapter 5] [40, Chapter 2]. However, the dierentiability results of a shape functional F : Ω → F (Ω) are usually stated and proved in terms of directional derivatives t ∈ R → F [(I + tθ)(Ω)] rather than Fréchet dierential.

Z[θ, θ] := II Ω θ∂Ω , θ ∂Ω + ∇ ∂Ω (θ n ) | θ∂Ω + ∇ ∂Ω ( θn ) | θ ∂Ω , (4)
Indeed, we usually have F (Ω) := Ω f , where the integrand f can depend on Ω, for example through the solutions of partial dierential equations (PDEs), or can only be dened on Ω, making diculties in dening f on the domain perturbations. Hence, in this case, it is easier to handle a real variable t than a space of vector elds θ. Since the two viewpoints are not entirely equivalent, we emphasize that we consider here the Fréchet setting for the derivatives.

Moreover, the shape derivatives of order higher than two are little studied [22, Section 5.9.7] although some structure theorems are available [START_REF] Novruzi | Structure of shape derivatives[END_REF] [22, Section 5.9.4] [START_REF] Bucur | Anatomy of the shape Hessian via lie brackets[END_REF]. Indeed, the second-order shape derivative is usually enough to conclude about the optimality of a shape [START_REF] Dambrine | On variations of the shape Hessian and sucient conditions for the stability of critical shapes[END_REF][START_REF] Dambrine | About stability of equilibrium shapes[END_REF], and even in this case, the theoretical/numerical computation are dicult, especially when PDEs are involved.

In our situation, things are much simpler because the integrand f is dened on the whole space and does not depend on the domain Ω. Therefore, our third main contribution in this paper is stated in Theorem 3.2. We prove that for any integer k 0, if f ∈ W k,1 and if Ω is measurable, then the associated map

F Ω : θ ∈ W 1,∞ → (I+θ)(Ω) f is of class C k around the origin.
The main achievement of Theorem 3.2 is to obtain an explicit formula (43) for the k-th-order shape derivative of F i.e. for the k-th-order dierential of F Ω at the origin for any integer k 1. In Theorem 3.3, we also manage to express (43) into a divergence form. Consequently, if Ω has a Lipschitz boundary, then the shape derivatives of F are expressed as boundary integrals (49) (50). In particular, in Corollary 3.5, we recover the well-known structure of shape gradient for the rst-order shape derivative of F :

∀θ ∈ W 1,∞ , D 0 F Ω (θ) = ∂Ω f θ n dA.
Furthermore, in Theorem 3.6, we show that for any integer k 2, if Ω is a C 1,1 -domain and if the perturbations are normal to the boundary (this hypothesis is fundamental here), then the k-th-order shape derivative of F has the following structure:

∀(θ 1 , . . . , θ k ) ∈ W 1,∞ ∩ C 1 k , D k 0 F Ω θ 1 , . . . , θ k = ∂Ω ∂ k F ∂Ω k (x) k i=1 θ i n (x) dA (x) . ( 5
)
The well-dened map ∂ k F ∂Ω k : ∂Ω → R depends on f and Ω but not on the perturbations (θ 1 , . . . , θ k ). Hence, by analogy with the nite-dimensional case, it is called the k-th-order partial derivative of F with respect to the domain Ω. In addition, it has an explicit expression which is given by:

∂ k F ∂Ω k = n i1,...,i k-1 =1 k-1 l=0 I l ⊆ 1,k-1 card I l =l p∈S I l s (p) ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [n Ω ] ij j∈I l [D ∂Ω n Ω ] ij i p(j) , (6)
where S I l is the set of permutations on I l i.e. of the bijective maps from I l into I l , and where s : S I l → {-1, 1} denotes the signature associated with permutations. In order to get back to the usual case of permutations on 1, l , we recall that the signature of a permutation p ∈ S I l is dened as s(p) := s(p -1

I l • p • p I l )
, where p I l is the unique strictly increasing map from 1, l into I l . We also emphasize the fact that the boundary values of the partial derivatives of f have to be understood in the sense of trace. In particular, the map ( 6) is uniquely determined on ∂Ω up to a set of zero A(∂Ω ∩ •)-measure, and as a consequence, it is correct to speak about the partial derivatives of F with respect to the domain Ω.

The formula ( 6) is very well known for little value of k but to our knowledge, such a general expression is new in its generality. We refer to Corollary 3.7 for a precise statement concerning the case k = 2 and for practical purpose, we compute the rst partial derivatives:

           ∂F ∂Ω = f ∂ 2 F ∂Ω 2 = ∇f | n Ω + H Ω f ∂ 3 F ∂Ω 3 = Hess f (n Ω ) | n Ω + 2H Ω ∇f | n Ω + f H 2 Ω -trace D ∂Ω n 2 Ω , (7)
where H Ω := div ∂Ω n Ω is the scalar mean curvature associated with the C

(•) := div(•) -D(•)n Ω | n Ω = trace[D ∂Ω (•)]
denoting the tangential component of the divergence operator.

To conclude this introduction, let us now explain how the paper is organized. In Section 2, we obtain the shape derivatives of p ν : Ω → p ν (Ω) for general wave functions. First, we dene the probability as a shape functional in Section 2.1. In Section 2.2, we give the general dierentiability result for p ν while in Section 2.3 (respectively Section 2.4), we treat the specic case of the rst-(resp. second-)order shape derivative of p ν . Then, in Section 3, we study the shape derivatives of a volume integral F : Ω → Ω f . We treat the measurable case in Section 3.1, the Lipschitz regularity in Section 3.2 and the C 1,1 -domains in Section 3.3. Finally, Section 4 is an appendix that gathers all the material and the proofs of standard results needed throughout the article.

2 Shape dierentiability for general wave functions 2.1 On the expression of the probability for general wave functions Let n 2 be an integer henceforth set. In this article, we consider a quantum system of n electrons whose chemical state is assumed to be entirely characterized by a given well-dened wave function [32, Section 1.1.1]:

Ψ : R 3 × -1 2 , 1 2 n -→ C x 1 σ 1 , . . . , x n σ n -→ Ψ x 1 σ 1 , . . . , x n σ n , (8)
where x i and σ i respectively refer to the space and spin variables of the i-th electron, for any i ∈ 1, n . Since we are dealing with fermions, we assume the antisymmetry of the wave function [41, Section 2.1.3], and we set L 2 ((R 3 × {-1 2 , 1 2 }) n , C) as the complex separable Hilbert space of all possible quantum states. However, we do not impose here a unitary L 2 -norm condition on the wave function as it is often the case. In other words, we make the following hypothesis.

Assumption 2.1. The map Ψ is a skew-symmetric form i.e. for any (i, j) ∈ 1, n 2 such that i = j, and for any (σ i , σ j ) ∈ {-1 2 , 1 2 } 2 and any (x i , x j ) ∈ R 3 × R 3 , we have:

Ψ . . . , x i σ i , . . . , x j σ j , . . . = -Ψ . . . , x j σ j , . . . , x i σ i , . . . . (9)
Moreover, the map Ψ is measurable and square integrable i.e. for any

(σ 1 , . . . , σ n ) ∈ {-1 2 , 1 2 } n , the following map belongs to L 2 ((R 3 ) n , C): Ψ (σ1,...,σn) : R 3 × . . . × R 3 -→ C (x 1 , . . . , x n ) -→ Ψ x 1 σ 1 , . . . , x n σ n . (10)
Finally, in addition to be well dened and nite, we assume that the following normalizing constant is a positive quantity i.e. it is not equal to zero:

c 0 := (σ1,...,σn)∈{-1 2 , 1 2 } n (R 3 ) n Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n . (11)
Hence, assuming that the wave function Ψ given in (8) satises Assumption 2.1, we can now use the traditional probabilistic interpretation of the wave function [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF]Section 1.1.1] in order to dene the shape functional in which we will be interested throughout the article. Indeed, the probability to nd for any i ∈ 1, n the electron i of spin σ i in a domain Ω i is proportional to:

Ω1×...×Ωn Ψ (σ1,...,σn) 2 ,
where Ψ (σ1,...,σn) is dened by [START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF]. Since Ψ (σ1,...,σn) is measurable, the above quantity is well dened for any (Lebesgue) measurable subsets Ω 1 , . . . , Ω n of R 3 , and it is nite since Ψ (σ1,...,σn) is square integrable. In particular, the probability to nd n electrons in a measurable set Ω ⊆ R 3 , regardless of their spins, is proportional to:

(σ1,...,σn)∈{-1 2 , 1 2 } n Ω n Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n .
The constant of proportionality is determined by the fact that we expect to nd n electrons in the whole space R 3 with probability one. Hence, let c 0 > 0 be as in [START_REF] Chamorro | Electron probability distribution in AIM and ELF basins[END_REF] and the probability p n (Ω) to nd n electrons in a measurable subset Ω of R 3 is given by the following well-dened quantity:

p n (Ω) := 1 c 0 (σ1,...,σn)∈{-1 2 , 1 2 } n Ω n Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n . (12)
Similarly, the probability p 0 (Ω) to nd zero electron in a measurable subset Ω of R 3 is dened as:

p 0 (Ω) := p n R 3 \Ω . ( 13
)
We now set ν ∈ 1, n -1 and search for the probability to nd exactly ν electrons in Ω, where exactly means that the n -ν remaining ones are located in the complement R 3 \Ω. The associated event can be interpreted as the reunion of the events nding exactly electrons i 1 , . . . , i ν in Ω, taken among all the subsets {i 1 , . . . , i ν } of ν pairwise distinct elements of 1, n . Hence, for any subset I ν ⊂ 1, n of ν elements i.e. such that card I ν = ν, we introduce the set Ω Iν := n i=1 A i , where

A i = Ω if i ∈ I ν otherwise A i = R 3 \Ω.
Following the same arguments than for p n and p 0 , the probability p ν (Ω) to nd exactly ν electrons in a measurable subset Ω of R 3 is given by the following well-dened quantity:

p ν (Ω) := 1 c 0 (σ1,...,σn)∈{-1 2 , 1 2 } n Iν ⊂ 1,n card Iν =ν Ω Iν Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n . ( 14
)
Then, we observe that such a nite reunion is disjoint i.e. Ω Iν ∩ Ω Jν = ∅ if I ν = J ν so we have:

p ν (Ω) = 1 c 0 (σ1,...,σn)∈{-1 2 , 1 2 } n     Iν ⊂ 1,n card Iν =ν Ω Iν Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n     .
Since ( 9) is satised by Ψ, we get

Ω Iν |Ψ (σ1,...,σn) | 2 = Ω ν ×(R 3 \Ω) n-ν |Ψ (σ p Iν (1) ,...,σ p Iν (n) ) | 2
, where

p Iν : 1, n → 1, n is a bijective map satisfying p Iν ( 1, ν ) = I ν . A new summation on the spin variables σi := σ p Iν (i) yields to σ1,...,σn Ω Iν |Ψ (σ1,...,σn) | 2 = σ1,...,σn Ω ν ×(R 3 \Ω) n-ν |Ψ (σ1,...,σn) | 2 ,
which does not depend on I ν any longer. It can thus be removed from the corresponding sum for which we know that card{I ν ⊂ 1, n , card

I ν = ν} = n! ν!(n-ν)! := n ν .
We deduce that:

p ν (Ω) = 1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n Ω ν ×(R 3 \Ω) n-ν Ψ x 1 σ 1 , . . . , x n σ n 2 dx 1 . . . dx n . (15)
Although [START_REF] Dambrine | About stability of equilibrium shapes[END_REF] should be the original denition for p ν in the sense that it is clear with ( 14) that p ν ∈ [0, 1] as it is the case for ( 12) and ( 13), we will however use the more practical formula [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF] for p ν in the remaining part of the article. In other words, we are now in position to properly dene the shape functional in which we will be interested throughout the article.

Denition 2.2. Assume that the wave function Ψ given in [START_REF] Causà | The bond analysis techniques (ELF and maximum probability domains): application to a family of models relevant to bio-inorganic chemistry[END_REF] satises Assumption 2.1. Let M be the set of all (Lebesgue) measurable subsets of R 3 and c 0 > 0 as in [START_REF] Chamorro | Electron probability distribution in AIM and ELF basins[END_REF]. Then, for any ν ∈ 0, n , the following shape functional is a well-dened map:

p ν : M -→ [0, 1] Ω -→ p ν (Ω) ,
where the probability p ν (Ω) to nd exactly ν electrons in the domain Ω is well dened by [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF] if ν ∈ 1, n -1 , by [START_REF] Dalphin | Some characterizations of a uniform ball property[END_REF] if ν = n, and by [START_REF] Dambrine | On variations of the shape Hessian and sucient conditions for the stability of critical shapes[END_REF] if ν = 0.

Remark 2.3. From the convention A 0 × B = B × A 0 = B for any sets A and B, we can deduce (12)(13) from [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF] by setting ν = 0 or ν = n in [START_REF] Daudel | Sur la localisabilité des corpuscules dans les noyaux et les cortèges électroniques des atomes et des molécules[END_REF]. We will adopt this convention in the article, in order to simplify the proofs and not to have to treat specically the cases ν = 0 and ν = n.

Finally, we can state a rst result concerning the symmetry property of the probability.

Lemma 2.4. Let M be the class of all (Lebesgue) measurable subsets of R 3 . We assume that the wave function Ψ given in [START_REF] Causà | The bond analysis techniques (ELF and maximum probability domains): application to a family of models relevant to bio-inorganic chemistry[END_REF] satises Assumption 2.1. Then, the map

p ν : Ω ∈ M → p ν (Ω) ∈ [0, 1]
of Denition 2.2 is well dened and we have:

p n R 3 = p 0 (∅) = 1 and ∀Ω ∈ M, p ν (Ω) = p n-ν R 3 \Ω .
In particular, the whole space R 3 (respectively the empty set ∅) is optimal for p n (resp. for p 0 ). Moreover, if Ω * is optimal for p ν , then R 3 \Ω * is optimal for p n-ν i.e.

∃Ω * ∈ M, p ν (Ω * ) = max Ω∈M p ν (Ω) =⇒ p n-ν R 3 \Ω * = max Ω∈M p n-ν (Ω) .
In other words, the shape optimization problem (1) only needs to be studied for integers ν n+1 2

and it has an obvious global maximizer if ν = 0 or if ν = n.

Proof. Let M contain all the measurable subsets of R 3 . We assume that the wave function Ψ given in (8) satises Assumption 2.1. Hence, from the foregoing, the map p ν : Ω ∈ M → p ν (Ω) ∈ [0, 1] introduced in Denition 2.2 is well dened. First, we get p n (R 3 ) = p 0 (∅) = 1 if we consider (12) [START_REF] Dambrine | On variations of the shape Hessian and sucient conditions for the stability of critical shapes[END_REF] with Ω = R 3 and Ω = ∅. Then, let Ω ∈ M. From (15), we get:

p ν (Ω) = 1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n Ω ν ×(R 3 \Ω) n-ν |Ψ (σ1,...,σn) | 2 ,
where Ψ (σ1,...,σn) and c 0 > 0 are respectively dened by [START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF] and [START_REF] Chamorro | Electron probability distribution in AIM and ELF basins[END_REF]. Using the property (9) of the wave function, we deduce that

Ω ν ×(R 3 \Ω) n-ν |Ψ (σ1,...,σn) | 2 = (R 3 \Ω) n-ν ×Ω ν |Ψ (σν+1,...,σn,σ1,...,σν ) | 2 .
Observing that n ν := n! ν!(n-ν)! = n n-ν and re-indexing the summation on the spin variables σi := σ ν+i for any i ∈ 1, n -ν and σi := σ i-n+ν for any i ∈ n -ν + 1, n , we obtain:

p ν (Ω) = 1 c 0 n n -ν (σ1,...,σn)∈{-1 2 , 1 2 } n (R 3 \Ω) n-ν ×Ω ν |Ψ (σ1,...,σn) | 2 = p n-ν R 3 \Ω .
Finally, if we assume that there exists Ω * ∈ M such that p ν (Ω * ) = max Ω∈M p ν (Ω), then for any Ω ∈ M, we deduce from the previous symmetry property:

p n-ν (Ω) = p ν R 3 \Ω max A∈M p ν (A) = p ν (Ω * ) = p n-ν R 3 \Ω * .
Hence, we get p n-ν (R 3 \Ω * ) = max Ω∈M p n-ν (Ω), concluding the proof of Lemma 2.4.

On the shape derivatives of the probability for measurable domains

First, we recall some terminology about shape dierentiability. We refer to Section 1 for notation.

Denition 2.5. Assume that the following shape functional is a well-dened map for a certain class of admissible shapes:

F : Ω -→ F (Ω) .
By abuse of terminology, we say that F is shape dierentiable at Ω if the following associated functional is well dened around the origin and Fréchet dierentiable at the origin:

F Ω : θ -→ F Ω (θ) := F [(I + θ) (Ω)] .
If it is the case, then the dierential D 0 F Ω of the map F Ω at the origin is called the (rst-order) shape derivative of F at Ω. Similarly, for any integer k 2, if F Ω is (k -1) times dierentiable around the origin and k times dierentiable at the origin, then we say that F is k times shape dierentiable at Ω, and the k-th-order dierential D k 0 F Ω of the map F Ω at the origin is called the k-th-order shape derivative of F at Ω. Moreover, by analogy with the nite-dimensional case, assume that there exists a unique well-dened function f Ω : ∂Ω → R such that:

D 0 F Ω (θ) = ∂Ω f Ω θ n dA.
Then, the map f Ω , eventually depending on Ω (but not on θ), is denoted by abuse of notation ∂F ∂Ω and called the shape gradient of F at Ω. Similarly, assume that in addition to the existence of a shape gradient, there exists a unique well-dened function fΩ : ∂Ω → R such that:

D 2 0 F Ω (θ, θ) = ∂Ω fΩ θ n θn dA - ∂Ω ∂F ∂Ω Z[θ, θ]dA,
where Z[θ, θ] is dened by (4). The map fΩ , eventually depending on Ω (but not on (θ, θ)), is denoted by abuse of notation ∂ 2 F ∂Ω 2 and called the shape Hessian of F at Ω.

The shape gradient and Hessian form are expected from shape derivatives [START_REF] Novruzi | Structure of shape derivatives[END_REF] [22, Section 5.9] [6] but note that in Denition 2.5, we did not clearly specify on which spaces are dened F and F Ω because such a structure depends on the required regularity for the domain Ω and the vector eld θ. We also recall that Z[θ, θ] dened by (4) represents the contribution of the tangential components. In particular, Z[θ, θ] ≡ 0 if θ and θ are orthogonal to the boundary ∂Ω i.e. if θ ∂Ω = θ∂Ω = 0.

Before stating our main result concerning the shape dierentiability of p ν : Ω → p ν (Ω) i.e. the dierentials at the origin of the associated map p ν,Ω : θ → p ν [(I + θ)(Ω)], we study the continuity properties of p ν,Ω . We recall that M refers to the class of measurable subset of R 3 , that C 0,1 denotes the set of Lipschitz continuous vector elds θ : R 3 → R 3 , that W 1,∞ := L ∞ ∩ C 0,1 is the set of Lipschitz continuous bounded vector elds, and that B 0,1 := {θ ∈ C 0,1 , θ C 0,1 < 1} is the open unit ball of C 0,1 centred at the origin i.e. the space of Lipschitz contractions. Lemma 2.6. Let n 2 and ν ∈ 0, n . Assume that the wave function Ψ given by [START_REF] Causà | The bond analysis techniques (ELF and maximum probability domains): application to a family of models relevant to bio-inorganic chemistry[END_REF] satises Assumption 2.1. Then, the shape functional p ν : Ω ∈ M → p ν (Ω) of Denition 2.2 is well dened, and for any Ω ∈ M, the associated map p ν,Ω :

θ ∈ C 0,1 → p ν [(I + θ)(Ω)] is well dened on B 0,1 . Moreover, for any Ω ∈ M, the map p ν,Ω : θ ∈ W 1,∞ → p ν [(I +θ)(Ω)] is continuous on B 0,1 ∩W 1,∞ .
Proof. Let n 2 and ν ∈ 0, n . We aim to consider the probability as a volume integral on an higher-dimensional space. For this purpose, we need to keep track of the dimension of the space in which we are working. Hence, the notation are modied in this direction. For example, M 3 now refers to the set of Ω ⊆ R 3 measurable, B 0,1 3 to the set of Lipschitz contraction θ : R 3 → R 3 , etc. Let θ ∈ B 0,1 3 . From Proposition 4.1, the Lipschitz continuous map I 3 + θ : x ∈ R 3 → x + θ(x) ∈ R 3 is bijective and its inverse (I 3 + θ) -1 is also Lipschitz continuous. In particular, (I 3 + θ) -1 is a measurable map and for any Ω ∈ M 3 , we get

(I 3 + θ)(Ω) ∈ M 3 . Hence, for any Ω ∈ M 3 , the map p ν,Ω : θ ∈ C 0,1 3 → p ν [(I 3 + θ)(Ω)] is well dened on B 0,1
3 . We now study its continuity by introducing the following higher-dimensional version of p ν :

pν : M 3n -→ R Ω -→ pν ( Ω) := 1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n Ω |Ψ (σ1,...,σn) | 2 , (16)
where [START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF] denes Ψ (σ1,...,σn) . Since Ψ (σ1,...,σn) and Ω are measurable, the integral is well dened and it is nite since Ψ (σ1,...,σn) is square integrable. Hence, the map pν is well dened by (16) and we can thus apply Lemma 3.1 to pν . For any Ω ∈ M 3n , the map pν, Ω : θ ∈ C 0,1 3n → pν [(I 3n + θ)( Ω)] is well dened on B 0,1 3n and moreover, the following map is continuous on

B 0,1 3n ∩ W 1,∞ 3n : pν, Ω : W 1,∞ 3n -→ R θ -→ pν, Ω ( θ) := pν I 3n + θ ( Ω) . (17)
Then, we want to relate pν and p ν . For this purpose, we consider the following map:

f : W 1,∞ (R 3 , R 3 ) -→ W 1,∞ (R 3 ) n , (R 3 ) n (θ : x → θ(x)) -→ f (θ) := (x 1 , . . . , x n ) → (θ (x 1 ) , . . . , θ (x n )) , (18)
which is well dened and linear. It is also continuous since one can check by direct calculations:

∀θ ∈ W 1,∞ 3 , f (θ) W 1,∞ 3n √ n θ W 1,∞ 3 and f (θ) C 0,1 3n = θ C 0,1 3 . ( 19
)
Moreover, let Ω ∈ M 3 . Since the set Ω ν × (R 3 \Ω) n-ν belongs to M 3n , we get:

p ν (Ω) = pν [Ω ν × (R 3 \Ω) n-ν ] and p ν,Ω = pν,Ω ν ×(R 3 \Ω) n-ν • f. ( 20
)
Since f is continuous and pν,

Ω ν ×(R 3 \Ω) n-ν is continuous on B 0,1 3n ∩ W 1,∞ 3n , we deduce from (19)(20) that p ν,Ω is continuous on B 0,1 3 ∩ W 1,∞ 3
, concluding the proof of Lemma 2.6.

We are now in position to prove our main shape dierentiability result concerning the shape functional p ν : Ω → p ν (Ω) of Denition 2.2. A striking feature is that we are able to get an explicit formula [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF] for the shape derivative of p ν at any order. As for Lemma 2.6, the proof completely relies on the shape dierentiability results of Section 3 for a volume integral. We refer to Section 1 for the notation and we recall that H k denotes the usual Sobolev space of L 2 -maps whose partial derivatives (in the weak distributional sense) are also L 2 -functions up to the order k.

Theorem 2.7. Let n 2, ν ∈ 0, n , and k 0 1 be three integers. First, we assume that the wave function Ψ given by [START_REF] Causà | The bond analysis techniques (ELF and maximum probability domains): application to a family of models relevant to bio-inorganic chemistry[END_REF] satises Assumption 2.1. In particular, the shape functional p ν : Ω → p ν (Ω) of Denition 2.2 is well dened. Moreover, we assume that the map Ψ (σ1,...,σn) dened by [START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF] belongs to H k0 ((R 3 ) n , C) for any (σ 1 , . . . , σ n ) ∈ {- 1 2 , 1 2 } n . Then, for any Ω ∈ M, the associated map p ν,Ω : θ ∈ W 1,∞ → p ν [(I + θ)(Ω)] is k 0 times Fréchet dierentiable at the origin and for any k ∈ 1, k 0 , its dierential of order k at the origin is given by the following continuous symmetric k-linear form dened for any (θ 1 , . . . , θ k ) ∈ W 1,∞ × . . . × W 1,∞ by:

D k 0 p ν,Ω (θ 1 , . . . , θ k ) := 1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n n i1,...,i k =1 3 m1,...,m k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) Ω ν ×(R 3 \Ω) n-ν ∂ k-l |Ψ (σ1,...,σn) | 2 j∈ 1,k j / ∈I l ∂ x ij mj (x 1 , . . . , x n )     j∈ 1,k j / ∈I l θ j x ij mj       j∈I l I ij i p(j) D xi j θ j mj m p(j)   dx 1 . . . dx n , (21)
where I ij i p(j) = 1 if i j = i p(j) otherwise zero. In other words, the functional p ν of Denition 2.2 is k 0 times shape dierentiable at any measurable subset of R n , and its shape derivative of order k is given by [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF] for any k ∈ 1, k 0 . Moreover, the associated map 1 and for any k ∈ 1, k 0 , its k-th-order Fréchet dierential is well dened by the following continuous map:

p ν,Ω : θ ∈ W 1,∞ → p ν [(I + θ)(Ω)] is k 0 times continuously dierentiable at any point of W 1,∞ ∩ B 0,
D k • p ν,Ω : W 1,∞ ∩ B 0,1 -→ L k c W 1,∞ k , R θ 0 -→ (θ 1 , ..., θ k ) → D k 0 p ν,(I+θ0)(Ω) θ 1 • (I + θ 0 ) -1 , ..., θ k • (I + θ 0 ) -1 , (22)
where D k 0 p ν,(I+θ0)(Ω) is the k-th-order shape derivative of p ν at (I +θ 0 )(Ω) given by [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF], and where L k c refers to the class of continuous k-linear maps. Proof. We use the notation of Lemma 2.6. We aim to relate [START_REF] Evans | Measure theory and ne properties of functions[END_REF] with p ν in order to use the results of Section 3 available for volume integrals. Let n 2, ν ∈ 0, n , and k 0 1 be three integers. We assume that the wave function Ψ given by (8) satises Assumption 2.1. In particular, the shape functional p ν : Ω ∈ M 3 → p ν (Ω) of Denition 2.2 is well dened. Moreover, we assume that the map Ψ (σ1,...,σn) in [START_REF] Causà | Maximum probability domains in the solid-state structures of the elements: the diamond structure[END_REF] belongs to

H k0 ((R 3 ) n , C) for any (σ 1 , . . . , σ n ) ∈ {-1 2 , 1 2 } n . Note that H k0 -regularity is required on Ψ to get the k 0 -th-order shape dierentiability. First, let Ω ∈ M 3 . From Lemma 2.6, the map p ν,Ω : θ ∈ W 1,∞ 3 ∩ B 0,1 3 → p ν [(I 3 + θ)(Ω)
] is well dened and continuous. We now show that it is k 0 times dierentiable at the origin. In the proof of Lemma 2.6, we also established that the map pν : Ω ∈ M 3n → pν ( Ω) is well dened by ( 16). Since we have |Ψ (σ1,...,σn) | 2 ∈ W k0,1 , we can apply Theorem 3.2 to the map pν , from which we deduce that pν,

Ω : θ ∈ W 1,∞ 3n → pν [(I 3n + θ)( Ω)
] is well dened and k 0 times dierentiable at the origin. Moreover, for any k ∈ 1, k 0 , its k-th-order dierential at the origin is given by the following continuous symmetric k-linear form:

∀( θ1 , . . . , θk ) ∈ W 1,∞ k , D k 0 pν, Ω ( θ1 , . . . , θk ) := 1 c 0 n ν n i1,...,i k =1 3 m1,...,m k =1 k l=0 I l ⊆ 1,k card I l =l (σ1,...,σn)∈{-1 2 , 1 2 } n Ω ∂ k-l |Ψ (σ1,...,σn) | 2 j∈ 1,k j / ∈I l ∂ x ij mj (x 1 , . . . , x n )     j∈ 1,k j / ∈I l θj (x 1 , . . . , x n ) 3(ij -1)+mj        p∈S I l s (p) j∈I l   ∂( θj ) ∂ x i p(j) m p(j) (x 1 , . . . , x n )   3(ij -1)+mj    dx 1 . . . dx n . (23)
Then, we want to relate the k-th-order shape derivative of pν with the one of p ν . Let Ω ∈ M 3 . We thus have Ω ν × (R 3 \Ω) n-ν ∈ M 3n . Considering the continuous linear map f given in [START_REF] Federer | Curvature measures[END_REF], we deduce from (19)(20) that the map p ν,Ω = pν,Ω ν ×(R 3 \Ω) n-ν • f is k 0 times dierentiable at the origin and for any k ∈ 1, k 0 , its k-th-order dierential at the origin is given by the following continuous symmetric k-linear form:

∀(θ 1 , . . . , θ k ) ∈ W 1,∞ 3 k , D k 0 p ν,Ω (θ 1 , . . . , θ k ) = D k 0 pν,Ω ν ×(R 3 \Ω) n-ν • f (θ 1 , . . . , θ k ) = D k f (0) pν,Ω ν ×(R 3 \Ω) n-ν [D 0 f (θ 1 ), . . . , D 0 f (θ k )] = D k 0 pν,Ω ν ×(R 3 \Ω) n-ν [f (θ 1 ), . . . , f (θ k )] . Since [f (θ j )(x 1 , . . . , x n )] 3(ij -1)+mj = [θ j (x ij )] mj , we deduce that [∂ (xi p(j) )m p(j) f (θ j )] 3(ij -1)+mj is equal to zero if i p(j) = i j otherwise it is equal to [∂ m p(j) θ j (x ij )] mj .
Using this observation in [START_REF] Lewis | The atom and the molecule[END_REF], we conclude that relation [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF] holds true. Let us now consider the second part of Theorem 3.2.

For any set Ω ∈ M 3n , the map pν,

Ω : θ ∈ W 1,∞ 3n → pν [(I 3n + θ)( Ω)
] is well dened and k 0 times continuously dierentiable at any point of W 1,∞ 3n ∩B 0,1 3n . Moreover, for any k ∈ 1, k 0 , its k-th-order dierential is well dened by the following continuous map:

D k • pν, Ω : W 1,∞ 3n ∩ B 0,1 3n -→ L k c W 1,∞ 3n k , R θ0 -→ D k θ0 pν, Ω := ( θ1 , . . . , θk ) → D k 0 pν,(I3n+ θ0)( Ω) θ1 • (I 3n + θ0 ) -1 , . . . , θk • (I 3n + θ0 ) -1 , (24)
where D k 0 pν,(I3n+ θ0)( Ω) is the k-th-order shape derivative of pν at (I 3n + θ0 )( Ω) given by [START_REF] Lewis | The atom and the molecule[END_REF]. Then, as before, we can relate the k-th-order dierential of pν, Ω with the one of p ν,Ω . Let Ω ∈ M 3 so we have Ω ν × (R 3 \Ω) n-ν ∈ M 3n . Considering the continuous linear map f given in [START_REF] Federer | Curvature measures[END_REF], we deduce from [START_REF] Francisco | Electron number probability distributions for correlated wave functions[END_REF] [START_REF] Gallegos | Maximal probability domains in linear molecules[END_REF]

that the map p ν,Ω = pν,Ω ν ×(R 3 \Ω) n-ν •f is k 0 times continuously dierentiable at any point θ 0 ∈ W 1,∞ 3 ∩B 0,1
3 and we have for any k ∈ 1, k 0 and for any (θ 1 , . . . , θ k ) ∈ W 1,∞ ×. . .×W 1,∞ :

D k θ0 p ν,Ω (θ 1 , . . . θ k ) = D k θ0 pν,Ω ν ×(R 3 \Ω) n-ν • f (θ 1 , . . . , θ k ) = D k f (θ0) pν,Ω ν ×(R 3 \Ω) n-ν [D θ0 f (θ 1 ) , . . . , D θ0 f (θ k )] = D k f (θ0) pν,Ω ν ×(R 3 \Ω) n-ν [f (θ 1 ) , . . . , f (θ k )] = D k 0 pν,[I3n+f(θ0)](Ω ν ×(R 3 \Ω) n-ν ) f (θ 1 ) • (I 3n + f (θ 0 )) -1 , . . . , f (θ k ) • (I 3n + f (θ 0 )) -1 ,
where we have used [START_REF] Lopes | Understanding maximum probability domains with simple models[END_REF] to obtain the last equality. Note also that using Proposition 4.1, we have

[I 3n + f (θ 0 )](Ω ν × (R 3 \Ω) n-ν ) = [(I 3 + θ 0 )(Ω)] ν × [R 3 \(I 3 + θ 0 )(Ω)] n-ν . We can also check that f (θ i ) • [I 3n + f (θ 0 )] -1 = f [θ i • (I 3 + θ 0 ) -1 ]
for any i ∈ 1, k so we deduce that:

D k θ0 p ν,Ω (θ 1 , . . . , θ k ) = D k 0 pν,[(I3+θ0)(Ω)] ν ×[R 3 \(I3+θ0)(Ω)] n-ν f θ 1 • (I 3 + θ 0 ) -1 , . . . , f θ k • (I 3 + θ 0 ) -1 = D k f (0) pν,[(I3+θ0)(Ω)] ν ×[R 3 \(I3+θ0)(Ω)] n-ν D 0 f θ 1 • (I 3 + θ 0 ) -1 , . . . , D 0 f θ k • (I 3 + θ 0 ) -1 = D k 0 pν,[(I3+θ0)(Ω)] ν ×[R 3 \(I3+θ0)(Ω)] n-ν • f θ 1 • (I 3 + θ 0 ) -1 , . . . , θ k • (I 3 + θ 0 ) -1 = D k 0 p ν,(I3+θ0)(Ω) θ 1 • (I 3 + θ 0 ) -1 , . . . , θ k • (I 3 + θ 0 ) -1 .
Hence, we have proved that the k-th-order dierential of p ν,Ω is well dened by the continuous map [START_REF] Henrot | Variation et optimisation de formes: une analyse géométrique[END_REF] for any k ∈ 1, k 0 , concluding the proof of Theorem 2.7.

On the rst-order shape derivative of the probability

We refer to Sections 1 and 2.12.2 for notation, especially Denition 2.5 for explanations about the notion of shape dierentiability. Theorem 2.7 is stated in the specic k 0 = 1 and we show that we can recover the shape gradient structure (2) by assuming the Lipschitz boundary of the domain.

Theorem 2.8. Let us consider the assumptions of Theorem 2.7 in the specic case k 0 = 1. Then, the following map is well dened and integrable:

P : R 3 × . . . × R 3 -→ R (x 1 , . . . , x n ) -→ P (x 1 , . . . , x n ) := ∇ x1 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x 1 ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) div θ (x 1 ) .
Moreover, the map p ν,Ω :

θ ∈ W 1,∞ → p ν [(I + θ)(Ω)]
is Fréchet dierentiable at the origin and its dierential is given by the following continuous linear form dened for any θ ∈ W 1,∞ by:

D 0 p ν,Ω (θ) = 1 c 0 n ν (σ1,...,σn)∈{ 1 2 , 1 2 } n ν Ω Ω ν-1 ×(R 3 \Ω) n-ν P (x 1 , . . . , x n ) dx 2 . . . dx n dx 1 + (n -ν) R 3 \Ω   Ω ν ×(R 3 \Ω) n-ν-1 P (x 1 , . . . , x n ) dx 2 . . . dx n   dx 1 . (25)
In other words, the functional p ν : Ω → p ν (Ω) of Denition 2.2 is shape dierentiable at any measurable subset of R 3 . If in addition, we now assume that Ω is an open bounded subset of R 3 with a Lipschitz boundary, then the shape derivative of p ν at Ω takes the form given in (2), where the shape gradient is uniquely determined up to a set of zero A(• ∩ ∂Ω)-measure, and dened for any point x ∈ ∂Ω by:

∂p ν ∂Ω (x) := 1 c 0 n ν (σ1,...,σn)∈{ 1 2 , 1 2 } n ν Ω ν-1 ×(R 3 \Ω) n-ν |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n )dx 2 . . . dx n -(n -ν) Ω ν ×(R 3 \Ω) n-ν-1 |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n ) dx 2 . . . dx n . (26)
In [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF], the boundary values of Ψ (σ1,...,σn) ∈ H 1 ((R 3 ) n , C) are understood in the sense of trace. Finally, the conventions

A 0 × B = B × A 0 = A, A -1 × B = B × A -1 = ∅, and ∅ f (x, y)dy = f (x) are used to interpret (2) and (25)(26) if ν ∈ {0, 1, n -1, n}.
Proof. First, the map P of the statement is well dened and integrable because θ ∈ W 1,∞ , Ψ ∈ H 1 , and ∇(|Ψ| 2 ) = 2Real(Ψ∇Ψ). Then, we can apply Theorem 2.7 with k 0 = k = 1. The functional p ν : Ω → p ν (Ω) of Denition 2.2 is thus shape dierentiable at Ω and its shape derivative D 0 p ν,Ω (θ) is dened for any θ ∈ W 1,∞ by the following quantity:

1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n Ω ν ×(R 3 \Ω) n-ν n i=1 ∇ xi |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x i ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) div θ (x i ) dx 1 . . . dx n . (27)
Finally, we can use the alternating property (9) satised by the wave function Ψ in order to get for any i ∈ 1, n , for any (x 1 , . . . ,

x n ) ∈ R 3 × . . . × R 3 , and for any (σ 1 , . . . , σ n ) ∈ {-1 2 , 1 2 } n : ∇ xi |Ψ (σ1,...,σi,...,σn) | 2 (x 1 , . . . , x i , . . . , x n ) = ∇ x1 |Ψ (σi,...,σ1,...,σn) | 2 (x i , . . . , x 1 , . . . , x n ) . ( 28
)
Inserting ( 28) in ( 27) and rearranging the summation on the spins variables in [START_REF] Montiel | Curves and surfaces[END_REF] by setting σi = σ 1 , σ1 = σ i , and σj = σ j for any j ∈ 1, n \{1, i}, we obtain that (25) holds true. It remains to study the Lipschitz case. Hence, we now assume that Ω is an open bounded subset of R 3 with a Lipschitz boundary. First, we recall that for any measurable subset A of (R 3 ) n-1 and for any g ∈ W 1,1 ((R 3 ) n , C), we have in the sense of distributions thus for almost every x ∈ R 3 :

A ∇ x1 (g)(x, x 2 , . . . , x n )dx 2 . . . dx n = ∇ x1 A g (•, x 2 , . . . , x n ) dx 2 . . . dx n (x) . (29)
Then, we can apply the Trace Theorem [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 4.3] in [START_REF] Menéndez | On the stability of some analytically solvable maximum probability domains[END_REF] for any θ ∈ W 1,∞ ∩ C 1 . Observing that the unit outer normal to the boundary ∂Ω = ∂(R 3 \Ω) satises n R 3 \Ω = -n Ω , we deduce that (2) holds true for any θ ∈ W 1,∞ ∩ C 1 . Finally, we can extend the result to any θ ∈ W 1,∞ from standard approximating arguments. Indeed, for any θ ∈ W 1,∞ , there exists a sequence

(θ i ) i∈N ⊂ W 1,∞ ∩ C 1 converging to θ strongly in L ∞ , weakly-star in W 1,∞
, and uniformly on compact sets (consider the usual mollier [17, Section 4.2.1 Theorem 1]). Note also that in [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF] the boundary values of Ψ (σ1,...,σn) ∈ H 1 ((R 3 ) n , C) are understood in the sense of trace. In particular, they are uniquely determined up to a set of zero A(• ∩ ∂Ω)-measure. It implies that the shape gradient of p ν is unique and well dened by [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF], concluding the proof of Theorem 2.8.

We conclude this section by specifying the Fréchet dierentiability property of the associated map p ν,Ω in the specic case k 0 = k = 1 of Theorem 2.7.

Corollary 2.9. Let us consider the assumptions of Theorem 2.7 in the specic case k 0 = 1.

Then, the map p ν,Ω :

θ ∈ W 1,∞ → p ν [(I + θ)(Ω)]
is well dened and continuously dierentiable at any point of W 1,∞ ∩ B 0,1 . Moreover, its (rst-order) dierential is well dened by the following continuous map:

D • p ν,Ω : W 1,∞ ∩ B 0,1 -→ L c W 1,∞ , R θ 0 -→ D θ0 p ν,Ω := θ → D 0 p ν,(I+θ0)(Ω) θ • (I + θ 0 ) -1 , (30)
where D 0 p ν,(I+θ0)(Ω) is the shape derivative of p ν at (I + θ 0 )(Ω) dened by [START_REF] Menéndez | On the stability of some analytically solvable maximum probability domains[END_REF]. If in addition, we assume that Ω is an open bounded subset of R 3 with a Lipschitz boundary, then the same result still holds true but we can now use the expression (2) to dene D 0 p ν,(I+θ0)(Ω) in [START_REF] Pendás | Pauling resonant structures in real space through electron number probability distributions[END_REF].

Proof. First, for measurable Ω ⊆ R 3 , the above statement is precisely the content of Theorem 2.7 with

k 0 = k = 1. If in addition, Ω is an open bounded subset of R 3 with a Lipschitz boundary, then (I + θ 0 )(Ω) is also a open bounded Lipschitz domain satisfying ∂[(I + θ 0 )(Ω)] = (I + θ 0 )(∂Ω). Moreover, θ • (I + θ 0 ) -1 ∈ W 1,∞ for any θ ∈ W 1,∞
and any θ 0 ∈ W 1,∞ ∩ B 0,1 so we deduce that the expression (2) denes well D 0 p ν,(I+θ0)(Ω) in [START_REF] Pendás | Pauling resonant structures in real space through electron number probability distributions[END_REF], concluding the proof of Corollary 2.9.

On the second-order shape derivative of the probability

We refer to Sections 1 and 2.12.2 for notation, especially Denition 2.5 for explanations about the notion of shape dierentiability. Theorem 2.7 is stated in the specic k 0 = 2 and we show that we can recover the shape Hessian structure (3) by assuming the C 1,1 -regularity of the domain.

Theorem 2.10. Let us consider the assumptions of Theorem 2.7 in the specic case k 0 = 2. First, the two following maps are well dened and integrable:

Q : R 3 × . . . × R 3 -→ R (x 1 , . . . , x n ) -→ Q (x 1 , . . . , x n ) := Hess x1 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) θ (x 1 ) | θ (x 1 ) + ∇ x1 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x 1 ) div θ (x 1 ) + θ (x 1 ) div θ (x 1 ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) div θ (x 1 ) div θ (x 1 ) -trace D x1 θD x1 θ R : R 3 × . . . × R 3 -→ R (x 1 , . . . , x n ) -→ R (x 1 , . . . , x n ) := |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) div θ (x 1 ) div θ (x 2 ) + ∇ x1 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x 1 ) div θ (x 2 ) + ∇ x2 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x 2 ) div θ (x 1 ) + 3 k,l=1 ∂ 2 (x1) k ,(x2) l |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) θ k (x 1 ) θl (x 2 ) .
We also introduce the following map, which is well dened and integrable if Ω is an open bounded subset of R 3 with a Lipschitz boundary and if

θ ∈ W 1,∞ ∩ C 1 : S : ∂Ω × R 3 × . . . × R 3 -→ R (x 1 , . . . , x n ) -→ S (x 1 , . . . , x n ) := ∇ x1 |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) | θ (x 1 ) θn (x 1 ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) div θ (x 1 ) θ (x 1 ) -D x1 θ θ (x 1 ) | n Ω (x 1 ) . (31)
Then, the map [START_REF] Pendás | Pauling resonant structures in real space through electron number probability distributions[END_REF] is Fréchet dierentiable at the origin i.e. p ν,Ω :

θ ∈ W 1,∞ → p ν [(I + θ)(Ω)]
is twice dierentiable at the origin and its second-order dierential is given by the following continuous symmetric bilinear form dened for any (θ, θ) ∈ W 1,∞ × W 1,∞ by:

D 2 0 p ν,Ω (θ, θ) = 1 c 0 n ν (σ1,...,σn)∈{ 1 2 , 1 2 } n ν Ω Ω ν-1 ×(R 3 \Ω) n-ν Q (x 1 , . . . , x n ) dx 2 . . . dx n dx 1 + (n -ν) R 3 \Ω Ω ν ×(R 3 \Ω) n-ν-1 Q (x 1 , . . . , x n ) dx 2 . . . dx n dx 1 + ν (ν -1) Ω×Ω Ω ν-2 ×(R 3 \Ω) n-ν R (x 1 , . . . , x n ) dx 3 . . . dx n dx 1 dx 2 + 2ν (n -ν) Ω×(R 3 \Ω) Ω ν-1 ×(R 3 \Ω) n-ν-1 R (x 1 , . . . , x n ) dx 3 . . . dx n dx 1 dx 2 + (n -ν) (n -ν -1) (R 3 \Ω)×(R 3 \Ω) Ω ν ×(R 3 \Ω) n-ν-2 R (x 1 , . . . , x n ) dx 3 . . . dx n dx 1 dx 2 .
(32) In other words, the functional p ν : Ω → p ν (Ω) of Denition 2.2 is twice shape dierentiable at any measurable subset of R 3 . If in addition, we now assume that Ω is an open bounded subset Ω of R 3 with a Lipschitz boundary, then the restriction

D • p ν,Ω : W 1,∞ ∩ C 1 ∩ B 0,1 → L c (W 1,∞ ∩ C 1 , R) remains dierentiable at the origin i.e. p ν,Ω : θ ∈ W 1,∞ ∩C 1 → p ν [(I +θ)(Ω)]
is twice dierentiable at the origin but its second-order dierential can now be dened for any (θ, θ) ∈ (W 1,∞ ∩ C 1 ) 2 by:

D 2 0 p ν,Ω (θ, θ) = ∂Ω×∂Ω K Ω (x, y) θ n (x) θn (y) dA (x) dA (y) + 1 c 0 n ν (σ1,...,σn)∈{ 1 2 , 1 2 } n ∂Ω ν Ω ν-1 ×(R 3 \Ω) n-ν S (x 1 , . . . , x n ) dx 2 . . . dx n -(n -ν) Ω ν ×(R 3 \Ω) n-ν-1 S (x 1 , . . . , x n ) dx 2 . . . dx n dA (x 1 ) , (33)
where the kernel K Ω : ∂Ω × ∂Ω → R is given for any (x, y) ∈ ∂Ω × ∂Ω by the following formula:

K Ω (x, y) = 1 c 0 n ν ν (ν -1) (σ1,...,σn)∈{ 1 2 , 1 2 } n Ω ν-2 ×(R 3 \Ω) n-ν |Ψ (σ1,...,σn) | 2 (x, y, x 3 , . . . , x n ) dx 3 . . . dx n -2ν (n -ν) Ω ν-1 ×(R 3 \Ω) n-ν-1 |Ψ (σ1,...,σn) | 2 (x, y, x 3 , . . . , x n ) dx 3 . . . dx n + (n -ν) (n -ν -1) Ω ν ×(R 3 \Ω) n-ν-2
|Ψ (σ1,...,σn) | 2 (x, y, x 3 , . . . , x n ) dx 3 . . . dx n .

(34) Finally, if we now assume that Ω is an open bounded subset of R 3 with a boundary of class C 1,1 , then the second-order shape derivative of p ν at Ω can take the form given by (3), where the shape Hessian is uniquely determined up to a set of zero A(• ∩ ∂Ω)-measure, and dened for any x ∈ ∂Ω by:

∂ 2 p ν ∂Ω 2 (x) = 1 c 0 n ν (σ1,...,σn)∈{ 1 2 , 1 2 } n ν Ω ν-1 ×(R 3 \Ω) n-ν H Ω (x) |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n ) + ∇ x1 |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n ) | n Ω (x) dx 2 . . . dx n -(n -ν) Ω ν ×(R 3 \Ω) n-ν-1 H Ω (x) |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n ) + ∇ x1 |Ψ (σ1,...,σn) | 2 (x, x 2 , . . . , x n ) | n Ω (x) dx 2 . . . dx n . (35)
In [START_REF] Politzer | Separation of core and valence regions in atoms[END_REF] and [START_REF] Savin | Probability distributions and valence shells in atoms[END_REF] [START_REF] Savin | On the signicance of ELF basins[END_REF], the boundary values of Ψ (σ1,...,σn) ∈ H 2 ((R 3 ) n , C) and ∇ x1 Ψ (σ1,...,σn) are understood in the sense of trace. The conventions

A 0 × B = B × A 0 = A, A -1 × B = B × A -1 = ∅, A -2 × B = B × A -2 = ∅, and ∅ f (x, y)dy = f (x) are used to interpret (3) and (32)(35) if n ∈ {2, 3} and ν ∈ {0, 1, 2, n -1, n -2, n}.
Proof. First, the maps Q and R of the statement are well dened and integrable because we have

θ, θ ∈ W 1,∞ , Ψ ∈ H 2 , ∇(|Ψ| 2 ) = 2Real(Ψ∇Ψ), and Hess(|Ψ| 2 ) = 2Real(ΨHessΨ + ∇Ψ[∇Ψ] T ).
The map S is also well dened and integrable but we have to assume the Lipschitz regularity of the domain Ω to get the existence almost everywhere of the unit normal eld n Ω , and also impose that θ ∈ W 1,∞ ∩ C 1 since we need to compute the boundary values of D • θ and div θ. Then, we can apply Theorem 2.7 with k 0 = k = 2. The functional p ν : Ω → p ν (Ω) of Denition 2.2 is thus twice shape dierentiable at Ω and its second-order shape derivative D 2 0 p ν,Ω (θ, θ) is dened for any (θ, θ) ∈ W 1,∞ × W 1,∞ by the following quantity:

1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n n i,j=1 3
k,l=1 Ω ν ×(R 3 \Ω) n-ν ∂ 2 |Ψ (σ1,...,σn) | 2 ∂(x i ) k ∂(x j ) l (x 1 , ..., x n ) [θ (x i )] k [ θ (x j )] l + ∂ |Ψ (σ1,...,σn) | 2 ∂ (x i ) k (x 1 , . . . , x n ) [θ (x i )] k [D xj θ] ll + ∂ |Ψ (σ1,...,σn) | 2 ∂ (x j ) l (x 1 , . . . , x n ) [ θ (x j )] l [D xi θ] kk + |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) [D xi θ] kk [D xj θ] ll -I ij [D xi θ] kl [D xj θ] lk .
Distinguishing the two cases i = j and i = j, we deduce that D 2 0 p ν,Ω (θ, θ) is equal to:

1 c 0 n ν (σ1,...,σn)∈{-1 2 , 1 2 } n n i=1 Ω ν ×(R 3 \Ω) n-ν   Hess xi |Ψ (σ1,...,σn) | 2 (x 1 , ..., x n )θ (x i ) | θ (x i ) + ∇ xi |Ψ (σ1,...,σn) | 2 (x 1 , . . . x n ) | θ (x i ) div θ (x i ) + θ (x i ) div θ (x i ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . x n ) div θ (x i ) div θ (x i ) -trace D xi θD xi θ + j∈ 1,n j =i 3 k,l=1 ∂ 2 |Ψ (σ1,...,σn) | 2 ∂(x i ) k ∂(x j ) l (x 1 , ..., x n ) [θ (x i )] k [ θ (x j )] l + ∇ xi |Ψ (σ1,...,σn) | 2 (x 1 , . . . x n ) | θ (x i ) div θ (x j ) + ∇ xj |Ψ (σ1,...,σn) | 2 (x 1 , . . . x n ) | θ (x j ) div θ (x i ) + |Ψ (σ1,...,σn) | 2 (x 1 , . . . x n ) div θ (x i ) div θ (x j )   dx 1 . . . dx n .
(36) We now proceed as in the proof of Theorem 2.8. We use the alternating property (9) of the wave function Ψ in order to get:

Hess xi |Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) = Hess x1 |Ψ (σi,...,σ1,...,σn) | 2 (x i , . . . , x 1 , . . . , x n ) and ∂ 2 |Ψ (σ1,...,σn) | 2 ∂(x i ) k ∂(x j ) l (x 1 , . . . , x n ) = ∂ 2 |Ψ (σi,σj ,...,σ1,...,σ2,...,σn) | 2 ∂(x 1 ) k ∂(x 2 ) l (x i , x j , . . . , x 1 , . . . , x 2 , . . . , x n ) .
Combining these observations and (28), we can now rearrange the summation on the spin variables in [START_REF] Scemama | Investigating the volume maximizing the probability of nding ν electrons from Variational Monte Carlo data[END_REF]. We deduce that relation [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF] holds true by distinguishing the cases

(x i , x j ) ∈ Ω × Ω, (x i , x j ) ∈ (R 3 \Ω) × (R 3 \Ω), (x i , x j ) ∈ Ω × (R 3 \Ω), and (x i , x j ) ∈ (R 3 \Ω) × Ω,
where the two last cases lead to the same expression by exchanging the role of i and j with (9) and relabelling again the spin variables.

Let us now study the Lipschitz case so we assume that Ω ⊂ R 3 is an open bounded set with a Lipschitz boundary and we consider the restriction map p ν,Ω :

θ ∈ W 1,∞ ∩ C 1 → p ν [(I + θ)(Ω)]. Since W 1,∞ ∩ C 1 is
also equipped with the W 1,∞ -norm, we deduce from Corollary 2.9 that p ν,Ω is continuously dierentiable on W 1,∞ ∩ C 1 ∩ B 0,1 , its dierential being well dened by:

D • p ν,Ω : W 1,∞ ∩ C 1 ∩ B 0,1 -→ L c W 1,∞ ∩ C 1 , R θ 0 -→ D θ0 p ν,Ω : θ → D 0 p ν,(I+θ0)(Ω) θ • (I + θ 0 ) -1 . (37)
Moreover, we obtain from the foregoing that the map ( 37) is dierentiable at the origin i.e. p ν,Ω is twice dierentiable at the origin and its second-order dierential is well dened by [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF] for any

(θ, θ) ∈ (W 1,∞ ∩ C 1 ) × (W 1,∞ ∩ C 1 ).
Let us now use the additional regularity of Ω, θ, and θ in order to improve the expression [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF]. On the one hand, one can observe that the map R of the statement can be expressed into a divergence form:

R (x 1 , . . . , x n ) = div θ ∇ x2 |Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) | θ (x 2 ) +|Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) div θ (x 2 ) (x 1 ) .
Arguing as in [START_REF] Pauling | The nature of the chemical bond: application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules[END_REF], we can thus apply the Trace Theorem [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 4.3] to the integrals involving R in [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF]. A similar argument with the variable x 2 yields to transform the integrals involving R in [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF] into the one involving the kernel K Ω in [START_REF] Rudin | Real and complex analysis[END_REF]. The sign obtained depends on the outer normal n R 3 \Ω = -n Ω of the boundary ∂Ω = ∂(R 3 \Ω). On the other hand, we can treat the integrals involving Q in (32) as follows. First, we assume that (θ, θ) ∈ W 2,∞ × W 2,∞ so we can write:

Q (x 1 , . . . , x n ) = div div |Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) θ θ -D • θ( θ) (x 1 ) = div div |Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) θ θ -D • θ(θ) (x 1 ) .
We emphasize that fact that the above equalities only hold true because we have assumed that

(θ, θ) ∈ W 2,∞ × W 2,∞ .
Then, arguing as in [START_REF] Pauling | The nature of the chemical bond: application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules[END_REF], we apply the Trace Theorem [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 4.3] for the integrals involving Q in [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF], from which we deduce the expressions involving S in [START_REF] Rudin | Real and complex analysis[END_REF].

More precisely, we need here to use the Trace Theorem for W 1,∞ -elds, which can be obtained from usual density arguments (see below [START_REF] Pauling | The nature of the chemical bond: application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules[END_REF]). Consequently, using again the fact that the unit outer normal to the boundary ∂Ω = ∂(R 3 \Ω) satises n R 3 \Ω = -n Ω , we have proved that [START_REF] Rudin | Real and complex analysis[END_REF] holds true for any (θ, θ) ∈ W 2,∞ × W 2,∞ . Note also that even if [START_REF] Rudin | Real and complex analysis[END_REF] is not symmetric in θ and θ, the symmetry can be obtained from the above equalities. Finally, we get that (33) holds true for any (θ, θ) It remains to treat the C 1,1 -regularity. First, we decompose the operators in S and the vector elds into a tangential and normal components. We thus have:

∈ (W 1,∞ ∩ C 1 ) × (W 1,∞ ∩ C 1 )
S (x 1 , . . . , x n ) = div ∂Ω |Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) θ (x 1 ) θn (x 1 ) + ∇ x1 |Ψ (x1,...,xn) | 2 (x 1 , . . . , x n ) | n Ω (x 1 ) θ n (x 1 ) θn (x 1 ) -|Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) D ∂Ω θ( θ∂Ω ) | n Ω (x 1 )
.

Then, we assume that Ω is an open bounded subset of R 3 with a boundary of class C

S (x 1 , . . . , x n ) = div ∂Ω |Ψ (σ1,...,σn) | 2 (•, x 2 , . . . , x n ) θn θ (x 1 ) + ∇ x1 |Ψ (x1,...,xn) | 2 (x 1 , . . . , x n ) | n Ω (x 1 ) θ n (x 1 ) θn (x 1 ) -|Ψ (σ1,...,σn) | 2 (x 1 , . . . , x n ) Z[θ, θ] (x 1 ) ,
where we recall that Z[θ, θ] is dened by (4). Finally, arguing as in [START_REF] Pauling | The nature of the chemical bond: application of results obtained from the Quantum Mechanics and from a theory of paramagnetic susceptibility to the structure of molecules[END_REF] with the above expression of S, we can apply the Divergence Theorem for surfaces [27, Theorem 6.10] in [START_REF] Rudin | Real and complex analysis[END_REF], which is valid with C 1,1 -regularity (adapt for example the proofs of [22, Proposition 5.4.9]). We deduce that the second-order shape derivative of p ν at Ω can take the form given in (3). We emphasize the fact that in [START_REF] Politzer | Separation of core and valence regions in atoms[END_REF] and (34) [START_REF] Savin | On the signicance of ELF basins[END_REF], the boundary values of Ψ (σ1,...,σn) ∈ H 2 ((R 3 ) n , C) and ∇ x1 Ψ (σ1,...,σn) have to be understood in the sense of trace. In particular, the shape Hessian ( 35) is uniquely determined up to a set of zero A(• ∩ ∂Ω)-measure and the same holds true for the kernel [START_REF] Savin | Probability distributions and valence shells in atoms[END_REF], concluding the proof of Theorem 2.10.

We conclude this section by specifying the Fréchet dierentiability of the associated map p ν,Ω in the specic case k 0 = k = 2 of Theorem 2.7.

Corollary 2.11. Let us consider the assumptions of Theorem 2.7 in the specic case k 0 = 2. Then, the map p ν,Ω : θ ∈ W 1,∞ → p ν [(I + θ)(Ω)] is well dened and twice continuously dierentiable at any point of W 1,∞ ∩ B 0,1 . Moreover, its second-order dierential is well dened by the following continuous map:

D 2 • p ν,Ω : W 1,∞ ∩ B 0,1 -→ L 2 c W 1,∞ × W 1,∞ , R θ 0 -→ (θ 1 , θ 2 ) → D 2 0 p ν,(I+θ0)(Ω) θ 1 • (I + θ 0 ) -1 , θ 2 • (I + θ 0 ) -1 , (38)
where D 2 0 p ν,(I+θ0)(Ω) is the second-order shape derivative of p ν at (I + θ 0 )(Ω) dened by [START_REF] Rivail | Eléments de chimie quantique à l'usage des chimistes[END_REF]. If in addition, we now assume that Ω is an open bounded subset of R 3 with a Lipschitz boundary, then the restriction map p ν,Ω : 1 and its second-order dierential is well dened by the following continuous map:

θ ∈ W 1,∞ ∩ C 1 → p ν [(I + θ)(Ω)] is still twice continuously dierentiable at any point of W 1,∞ ∩ C 1 ∩ B 0,
D 2 • p ν,Ω : W 1,∞ ∩ C 1 ∩ B 0,1 -→ L 2 c W 1,∞ ∩ C 1 × W 1,∞ ∩ C 1 , R θ 0 -→ (θ 1 , θ 2 ) → D 2 0 p ν,(I+θ0)(Ω) θ 1 • (I + θ 0 ) -1 , θ 2 • (I + θ 0 ) -1 , (39)
where D 2 0 p ν,(I+θ0)(Ω) can now be dened by [START_REF] Rudin | Real and complex analysis[END_REF]. Finally, if we assume that Ω is an open bounded subset of R 3 with a boundary of class C 1,1 , then the last result still holds true but we can now use the expression (3) to dene D 2 0 p ν,(I+θ0)(Ω) in [START_REF] Silvi | Classication of chemical bonds based on topological analysis of electron localization functions[END_REF]. Proof. First, for a measurable Ω ⊆ R 3 , the statement ( 38) is precisely the content of Theorem 2.7 with k 0 = k = 2. If in addition, Ω is an open bounded subset of R 3 with a Lipschitz boundary, then

(I 3 + θ 0 )(Ω) is also a open bounded Lipschitz domain satisfying ∂[(I 3 + θ 0 )(Ω)] = (I 3 + θ 0 )(∂Ω). Moreover, we have θ•(I 3 +θ 0 ) -1 ∈ W 1,∞ ∩C 1 for any θ ∈ W 1,∞ ∩C 1 and any θ 0 ∈ W 1,∞ ∩C 1 ∩B 0,1 .
We deduce that the expression (33) denes well D 2 0 p ν,(I+θ0)(Ω) in [START_REF] Silvi | Classication of chemical bonds based on topological analysis of electron localization functions[END_REF]. Finally, if Ω is now an open bounded subset of R 3 with a C 1,1 -boundary, then (I + θ 0 )(Ω) is also a C 1,1 -domain and we can use the expression (3) to dene D 2 0 p ν,(I+θ0)(Ω) in [START_REF] Silvi | Classication of chemical bonds based on topological analysis of electron localization functions[END_REF], concluding the proof of Corollary 2.11.

3 About the shape derivatives of a volume integral

In this section, the integer n 2 is still xed but now refers to the dimension of the real space R n in which we are working (and not to the number of electrons as it was the case in Sections 12).

In particular, Ω now denotes a subset of R n and θ : R n → R n is a well-dened vector eld on R n .

Our goal here is to study the shape dierentiability properties of the following map:

F : M -→ R Ω -→ F (Ω) := Ω f (x) dx, (40)
where the integration is done with respect to the n-dimensional Lebesgue measure, and where M now refers to the class of (Lebesgue) measurable subset of R n . We refer to Sections 1 and 2.12.2 for notation, especially Denition 2.5 for explanations about the notion of shape dierentiability. First, note that if f ∈ L 1 (R n , R), the map F : Ω → F (Ω) is well dened by [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF].

Then, we aim to establish precise shape dierentiability results concerning [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] since the proofs of Section 2 were all relying on the shape derivatives of a volume integral. We distinguish three cases according to the regularity of the given domain. We also mention that from our statements, we can recover the standard formulas for the rst-and second-order shape derivatives of a volume integral [START_REF] Delfour | Shapes and geometries: metrics, analysis, dierential calculus, and optimization[END_REF]Chapter 9]

The general case of a measurable domain

For any integer k 1, we dene W k,1 as the standard Sobolev space of L 1 -maps from R n into R whose partial derivatives (in the weak distributional sense) are also L 1 -functions up to the order k. In this section, we prove that if f ∈ W k, 1 and Ω ∈ M, then the map [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] is of class C k around the origin. In particular, we get an explicit formula (43) for the shape derivative of a volume integral at any order. The proof is made by induction on k so we rst need to initialize the process by studying the continuity properties of

F Ω : θ ∈ W 1,∞ → F [(I +θ)(Ω)] associated with
F Ω . Lemma 3.1. Let n 2, f ∈ L 1 (R n , R),
and Ω ∈ M. Then, the map

F Ω : θ ∈ C 0,1 → F [(I +θ)(Ω)] is well dened on B 0,1 . Moreover, F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is continuous on B 0,1 ∩ W 1,∞ . Proof. Let n 2, f ∈ L 1 (R n , R),
and Ω ∈ M. First, applying Proposition 4.1, the map I + θ has a Lipschitz continuous inverse for any θ ∈ B 0,1 , from which we deduce that (I + θ)(Ω) is measurable. Hence, the map

F Ω : θ ∈ C 0,1 → F [(I + θ)(Ω)] is well dened on B 0,1 . Then, let θ ∈ W 1,∞ ∩ C 0,1 .
We use the change of variables formula valid for any Lipschitz continuous map [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 3.3.3] and the (reverse) triangle inequality in order to get:

(I+θ)(Ω) f - Ω f Ω [f • (I + θ) -f ] |det [D • (I + θ)] | + Ω f (|det [D • (I + θ)] | -1) det [D • (I + θ)] L ∞ f • (I + θ) -f L 1 + det [D • (I + θ)] -1 L ∞ f L 1 .
Combining (63) and the continuity at the origin of the Jacobian determinant of (I + θ) ensured by Proposition 4.4 with the one of θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) ensured by Proposition 4.6, we can let θ W 1,∞ → 0 in the above inequality. Hence, the map

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)
] is continuous at the origin. Finally, let θ ∈ W 1,∞ ∩ B 0,1 . We recall that (I + θ)(Ω) is measurable and moreover, note that for any h ∈ W 1,∞ such that h W 1,∞ < 1 -θ C 0,1 , we have θ + h C 0,1 < 1 so we can write:

(I+θ+h)(Ω) f - (I+θ)(Ω) f = (I+h θ )(Ω θ ) f - Ω θ f, (41)
where we have set Ω θ := (I + θ)(Ω) and

h θ := h • (I + θ) -1 . One can check that h θ L ∞ h L ∞ and h θ C 0,1 h C 0,1 (I + θ) -1 C 0,1 h C 0,1 (1 -θ C 0,1
) -1 by Proposition 4.1 so we deduce that:

h θ W 1,∞ h L ∞ + h C 0,1 1 -θ C 0,1 = h W 1,∞ -h L ∞ θ C 0,1 1 -θ C 0,1 h W 1,∞ 1 -θ C 0,1 . (42)
In particular, we have

h θ W 1,∞ → 0 as h W 1,∞ → 0.
Considering the continuity at the origin of [START_REF] Szabo | Modern Quantum Chemistry: introduction to advanced electronic structure theory[END_REF]. We have obtained the continuity of

h θ ∈ W 1,∞ → F [(I + h θ )(Ω θ )], we can let h W 1,∞ → 0 in
F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] on W 1,∞ ∩ B 0,1
, concluding the proof of Lemma 3.1.

Theorem 3.2. Let n 2 and k 0 1 be two integers. We consider f ∈ W k0,1 (R n , R) and Ω ∈ M.

Then, the map

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)
] is k 0 times Fréchet dierentiable at the origin and for any k ∈ 1, k 0 , its dierential of order k at the origin is given by the following continuous symmetric k-linear form dened for any (θ 1 , . . . , θ k ) ∈ W 1,∞ × . . . × W 1,∞ by:

D k 0 F Ω (θ 1 , . . . , θ k ) = n i1,...,i k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) Ω ∂ k-l f j∈ 1,k j / ∈I l ∂x ij (x) j∈ 1,k j / ∈I l [θ j (x)] ij j∈I l [D x θ j ] ij i p(j) dx.
(43) In other words, the functional [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] is k 0 times shape dierentiable at any measurable subset of R n and its k-th-order shape derivative is well dened by (43) for any k ∈ 1, k 0 . Moreover, the map

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is k 0 times continuously dierentiable at any point of W 1,∞ ∩ B 0,1
and for any k ∈ 1, k 0 , its k-th-order dierential is well dened by the following continuous map:

D k • F Ω : W 1,∞ ∩ B 0,1 -→ L k c W 1,∞ k , R θ 0 -→ (θ 1 , ..., θ k ) → D k 0 F (I+θ0)(Ω) θ 1 • (I + θ 0 ) -1 , ..., θ k • (I + θ 0 ) -1 , (44) where D k 0 F (I+θ0)(Ω)
is the k-th-order shape derivative of F at (I + θ 0 )(Ω) given by (43). Proof. We are going to prove this result by induction on the integer k ∈ N. First, recalling the usual conventions ∂ 0 f = f , i∈∅ = 0, i∈∅ = 1, and D 0 θ0 F Ω = F Ω (θ 0 ) = F (I+θ0)(Ω) (0), we deduce from Lemma 3.1 that Theorem 3.2 holds true for k = 0. Let us assume that it is also true for some integer k 0. Let n 2 be an integer, f ∈ W k+1,1 , and Ω ∈ M. The induction hypothesis ensures that the map 1 . We now show that the additional regularity assumption we made on f allows the function (44) to be dierentiable at the origin. Let (θ 0 , θ 1 , . . . , θ k ) ∈ (W 1,∞ ) k+1 be such that θ 0 C 0,1 < 1. First, we express the k-th-order dierential in a simpler form, using the change of variables formula valid for Lipschitz continuous maps [17, Section 3.3.3]:

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is k times continuously dierentiable at any point of W 1,∞ ∩ B 0,
D k θ0 F Ω (θ 1 , . . . , θ k ) := n i1,...,i k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) (I+θ0)(Ω) ∂ k-l f j∈ 1,k j / ∈I l ∂x ij j∈ 1,k j / ∈I l θ j • (I + θ 0 ) -1 ij j∈I l D • θ j • (I + θ 0 ) -1 ij i p(j) = n i1,...,i k =1 k l=0 I l ⊆ 1,k cardI l =l p∈S I l s (p) Ω ∂ k-l f j∈ 1,k j / ∈I l ∂x ij • (I + θ 0 ) j∈ 1,k j / ∈I l [θ j ] ij j∈I l D (I+θ0)(•) θ j • (I + θ 0 ) -1 ij i p(j) |det[D • (I + θ 0 )]|.
For any θ ∈ W 1,∞ , we dene Def (θ) as the set of points in R n at which θ is dierentiable. Note that from Rademacher's Theorem [17, Section 3.1.2], R n \Def(θ) has a zero n-dimensional Lebesgue measure. We can now introduce the set

A := Def (θ) ∩ Def (θ 0 ) ∩ (I + θ 0 ) -1 [Def (θ • (I + θ 0 ) -1 )].
From Lemma 4.2, we get for almost every point x ∈ R n (more precisely for any x ∈ A since R n \A has a zero n-dimensional Lebesgue measure):

D x θ (I + D x θ 0 ) -1 = D (I+θ0)(x) θ • (I + θ 0 ) -1 .
Furthermore, we get from Proposition 4.

4 that det[D • (I+θ)] → 1 for the L ∞ -norm as θ W 1,∞ → 0.
Hence, there exists δ ∈]0, 1[ such that for any θ 0 W 1,∞ < δ, the Jacobian determinant of (I +θ 0 ) is positive. Combining these two observations, we obtain from the foregoing that for any θ 0 ∈ W 1,∞ such that θ 0 W 1,∞ < δ:

D k θ0 F Ω (θ 1 , . . . , θ k ) = n i1,...,i k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) Ω ∂ k-l f j∈ 1,k j / ∈I l ∂x ij • (I + θ 0 ) j∈ 1,k j / ∈I l [θ j ] ij j∈I l D • θ j (I + D • θ 0 ) -1 ij i p(j) det [D • (I + θ 0 )] .
(45) Then, we introduce the continuous (k+1)-linear form (43), which is symmetric i.e. for any p ∈ S k+1 and any (θ 1 , . . . , θ k+1 ) ∈ (W 1,∞ ) k+1 , we have D k+1 0 F Ω (θ p(1) , . . . , θ p(k+1) ) = D k+1 0 F Ω (θ 1 , . . . , θ k+1 ). We now prove that this good candidate is the (k + 1)-th order dierential of F Ω at the origin. For this purpose, we express it dierently. We emphasize the fact that we have not (yet) proved that (43) is the (k + 1)-th order dierential of F Ω but we use its notation for convenience. We set θ k+1 := θ 0 to keep this in mind. We thus have:

D k+1 0 F Ω (θ 1 , . . . , θ k , θ 0 ) := n i1,...,i k ,i k+1 =1 k+1 l=0 I l ⊆ 1,k+1 card I l =l p∈S I l s (p) Ω ∂ k+1-l f j∈ 1,k+1 j / ∈I l ∂x ij j∈ 1,k+1 j / ∈I l [θ j ] ij j∈I l [D • θ j ] ij i p(j) .
We split the above expression into two disjoint situations, the last one being itself splitted into two subcases. In the rst situation, we assume k + 1 / ∈ I l . In this particular case, the sum on l can stop at k since we are assuming that I l has at most k elements. Moreover, we can explicit the indice i k+1 and the subset I l is included in 1, k . In the second situation, we assume k + 1 ∈ I l . Similarly, the sum on l can start from one since we are assuming that I l is not empty. Then, two subcases follow. On the one hand, we assume p(k + 1) = k + 1. In this case, this is equivalent to search only for subsets I l-1 ⊆ 1, k of l -1 pairwise distinct elements, and also bijective maps q : I l-1 → I l-1 , then set I l := I l-1 ∪ {k + 1} and p := q on I l-1 with p(k + 1) := k + 1. On the other hand, we assume p(k + 1) = k + 1 so we can make a partition on the bijections p : I l → I l by xing the element k + 1. We thus have:

D k+1 0 F Ω (θ 1 , . . . , θ k , θ 0 ) = n i1,...,i k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) Ω n i k+1 =1 ∂ ∂x i k+1      ∂ k-l f j∈ 1,k j / ∈I l ∂x ij      [θ 0 ] i k+1 j∈ 1,k j / ∈I l [θ j ] ij j∈I l [D • θ j ] ij i p(j) + n i1,...,i k =1 k+1 l=1 I l-1 ⊆ 1,k card I l-1 =l-1 I l :=I l-1 ∪{k+1} q∈S I l-1 p:=q on I l-1 p(k+1):=k+1 s (p) Ω ∂ k-(l-1) f j∈ 1,k j / ∈I l-1 ∂x ij j∈ 1,k j / ∈I l-1 [θ j ] ij j∈I l-1 [D • θ j ] ij i q(j) n i k+1 =1 [D • θ 0 ] i k+1 i k+1 + n i1,...,i k =1 k+1 l=1 I l-1 ⊆ 1,k card I l-1 =l-1 I l :=I l-1 ∪{k+1} j0∈I l-1 p∈S I l p(k+1) =k+1 p(j0)=k+1 s(p) Ω ∂ k-(l-1) f j∈ 1,k j / ∈I l-1 ∂x ij j∈ 1,k j / ∈I l-1 [θ j ] ij j∈I l-1 j =j0 [D • θ j ] ij i p(j) n i k+1 =1 [D • θ j0 ] ij 0 i k+1 [D • θ 0 ] i k+1 i p(k+1) .
In the second integral above, p(k + 1) = k + 1 so the number of transpositions needed to decompose p -1

I l • p • p I l is the same than for p -1 I l-1 • q • p I l-1
, from which we deduce that s(p) = s(q). Moreover, in the last integral above, we make a change of indices r := p • t, where t is only exchanging k + 1 and j 0 . Since the signature is a morphism of group, we have s(r) = s(p • t) = s(p)s(t) = -s(p). Indeed, t permutes two indices of I l thus p -1 I l • t • p I l ∈ S l is a transposition, whose signature is -1. We are then back to a summation on r for which r(k + 1) = k + 1 i.e. in the previous situation of the second integral above. We can thus do the same foregoing procedure. We obtain:

D k+1 0 F Ω (θ 1 , . . . , θ k , θ 0 ) = n i1,...,i k =1 k l=0 I l ⊆ 1,k card I l =l p∈S I l s (p) Ω ∇ ∂ k-l f j∈ 1,k j / ∈I l ∂x ij | θ 0 j∈ 1,k j / ∈I l [θ j ] ij j∈I l [D • θ j ] ij i p(j) + n i1,...,i k =1 k+1 l=1 I l-1 ⊆ 1,k card I l-1 =l-1 I l :=I l-1 ∪{k+1} q∈S I l-1 p:=q on I l-1 p(k+1):=k+1 s (q) Ω ∂ k-(l-1) f j∈ 1,k j / ∈I l-1 ∂x ij j∈ 1,k j / ∈I l-1 [θ j ] ij j∈I l-1 [D • θ j ] ij i q(j) div (θ 0 ) + n i1,...,i k =1 k+1 l=1 I l-1 ⊆ 1,k card I l-1 =l-1 I l :=I l-1 ∪{k+1} j0∈I l-1 q∈S I l-1 r:=q on I l-1 r(k+1)=k+1 s (q) Ω ∂ k-(l-1) f j∈ 1,k j / ∈I l-1 ∂x ij j∈ 1,k j / ∈I l-1 [θ j ] ij [-D • θ j0 D • θ 0 ] ij 0 i q(j 0 ) j∈I l-1 j =j0 [D • θ j ] ij i q(j) .
Note that in the last product, we have replaced i r[t(j)] by i q(j) since they coincide on I l-1 \{j 0 }. Finally, we can notice that in the two last integrals, we have expressed everything in terms of I l-1 and q and so we can drop the notation I l , p, and r. Re-indexing the summation on l in the two last integrals by m = l -1, we get from all these observations:

D k+1 0 F Ω (θ 1 , . . . , θ k , θ 0 ) = n i1,...,i k =1 k m=0 Im⊆ 1,k card Im=m q∈S Im s (q) Ω j∈ 1,k j / ∈Im [θ j ] ij   ∇ ∂ k-m f j∈ 1,k j / ∈Im ∂x ij | θ 0 j∈Im [D • θ j ] ij i q(j) + ∂ k-m f j∈ 1,k j / ∈Im ∂x ij j∈Im [D • θ j ] ij i q(j) div (θ 0 ) + j0∈Im ∂ k-m f j∈ 1,k j / ∈Im ∂x ij j∈Im j =j0 [D • θ j ] ij i q(j) [-D • θ j0 D • θ 0 ] ij 0 i q(j 0 )   .
(46) We now introduce some more notation in order to handle the quantities (43)(46) we want to estimate. For this purpose, we set:

a 0 := ∂ k-m f j∈ 1,k j / ∈Im ∂x ij • (I + θ 0 ), b 0 := ∂ k-m f j∈ 1,k j / ∈Im ∂x ij , c 0 := ∇ ∂ k-m f j∈ 1,k j / ∈Im ∂x ij | θ 0 ,
and for any j ∈ 1, m :

     a j := D • θ p Im (j) (I + D • θ 0 ) -1 i p Im (j) i q[p Im (j)] b j := [D • θ p Im (j) ] i p Im (j) i q[p Im (j)] c j := -[D • θ p Im (j) D • θ 0 ] i p Im (j) i q[p Im (j)] .
We also set a m+1 := det[D • (I + θ 0 )], b m+1 := 1, and c m+1 := div(θ 0 ). Then, we introduce the following map:

R k (θ 0 , . . . , θ k ) := D k θ0 F Ω (θ 1 , . . . , θ k ) -D k 0 F Ω (θ 1 , . . . , θ k ) -D k+1 0 F Ω (θ 1 , . . . , θ k , θ 0 ) .
Considering the expressions (43) and the ones (45)(46) we have established, in each product/sum concerning j ∈ I m , we make a change of indices u := p -1 Im (j) so as to order the product/sum from u = 1 to u = m. We obtain with our notation:

R k (θ 0 , . . . , θ k ) = n i1,...,i k =1 k m=0 Im⊆ 1,k card Im=m q∈S Im s (q) Ω j∈ 1,k j / ∈Im [θ j ] ij     m+1 u=0 a u - m+1 u=0 b u - m+1 u0=0 c u0 u∈ 0,m+1 u =u0 b u     . = n i1,...,i k =1 k m=0 Im⊆ 1,k card Im=m q∈S Im s (q) Ω j∈ 1,k j / ∈Im [θ j ] ij m+1 u0=0 u0-1 u=0 b u (a u0 -b u0 -c u0 ) m+1 u=u0+1 a u + c u0 m+1 l=u0+1 l-1 u=u0+1 a u (a l -b l ) m+1 u=l+1 b u .
Therefore, we can now estimate each term in the last equality in order to obtain the required relation

|R k (θ 0 , . . . , θ k )| R(n, f, k, θ 0 ) k j=0 θ j W 1,∞ with |R(n, f, k, θ 0 )| → 0 as θ 0 W 1,∞ → 0.
Let us detail this procedure. First, we can apply Proposition 4.6 to the maps ∂ k-m f ∈ W 1,1 , m ∈ 0, k , then use the Cauchy-Schwarz inequality with (61), and combine the relations (62)(63) with the fact that θ 0 W 1,∞ < δ < 1. We deduce that:

(a 0 -b 0 -c 0 ) m u=1 a u a m+1 L 1 (n -1)! n 1 -δ m+1 θ 0 W 1,∞ R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 W 1,∞ → 0. Hence, we have estimated the rst term of the rst sum. We can proceed similarly for the other ones. Using the L 1 -norm for the maps ∂ k-m f ∈ W 1,1 , m ∈ 0, k , and the L ∞ -norm for the remaining terms, we get from the Cauchy-Schwarz inequality with (61), relations (62)(63) with θ 0 W 1,∞ < δ < 1, and Proposition 4.3:

m u0=1 b 0 u0-1 u=1 b u (a u0 -b u0 -c u0 ) m u=u0+1 a u a m+1 L 1 m √ n(n -1)! n 1 -δ m f W k,1 θ 0 C 0,1 R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. The same arguments and Proposition 4.4 also yield to:

b 0 m u=1 b u (a m+1 -b m+1 -c m+1 ) L 1 f W k,1 θ 0 C 0,1 R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. We next observe that m k and δ only depends on n (in fact one can prove that δ = 1 1+n! ), where we recall that δ ∈]0, 1[ is such that the Jacobian determinant of (I + θ) is positive for any θ W 1,∞ < δ. Gathering the three last estimations and these observations, we thus have obtained:

m+1 u0=0 u0-1 u=0 b u (a u0 -b u0 -c u0 ) m+1 u=u0+1 a u L 1 C (n, k, f ) θ 0 W 1,∞ R(θ 0 ) j∈Im θ j W 1,∞ , (47)
where C(n, k, f ) > 0 is a xed constant depending only on n, k, and f W k,1 , and where |R(θ 0 )| → 0 as θ 0 W 1,∞ → 0. We continue our estimations. Arguing as in (70) with θ 0 and ∂ k-m f ∈ W 1,1 , m ∈ 0, k , we use the Cauchy-Schwarz inequality with (61), relation (62) with θ 0 W 1,∞ < δ < 1, and Proposition 4.3 in order to get:

c 0 m l=1 l-1 u=1 a u (a l -b l ) m u=l+1 b u b m+1 L 1 m √ n n 1 -δ m-1 f W k+1,1 θ 0 L ∞ R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. The same arguments combined with Proposition 4.4 give:

c 0 m u=1 a u (a m+1 -b m+1 ) L 1 n 1 -δ m f W k+1,1 θ 0 L ∞ R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ C 0,1 → 0. Similarly, we get from Proposition 4.3:

m u0=1 b 0 u0-1 u=1 b u c u0 m l=u0+1 l-1 u=u0+1 a u (a l -b l ) m u=l+1 b u b m+1 L 1 n √ n n 1 -δ m-2 m(m -1) f W k,1 θ 0 C 0,1 R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0, and also from Proposition 4.4:

m u0=1 b 0 u0-1 u=1 b u c u0 m u=u0+1 a u (a m+1 -b m+1 ) L 1 nm n 1 -δ m-1 f W k,1 θ 0 C 0,1 R(θ 0 ) j∈Im θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. Gathering the four last estimations and observing again that m k and δ < 1 only depends on n, we obtain:

m+1 u0=0 u0-1 u=0 b u c u0 m+1 l=u0+1 l-1 u=u0+1 a u (a l -b l ) m+1 u=l+1 b u L 1 C(n, k, f W k+1,1 ) θ 0 W 1,∞ R(θ 0 ) j∈Im θ j W 1,∞ , (48)
where C(n, k, f W k+1,1 ) > 0 is a xed constant depending only on n, k, and f W k+1,1 , and where |R(θ 0 )| → 0 as θ C 0,1 → 0. Finally, we use (47)(48) to estimate the last expression obtained for R k (θ 0 , θ 1 , . . . , θ k ). We deduce that:

|||R k (θ 0 , •, . . . , •)||| := sup (θ1,...,θ k )∈(W 1,∞ ) k θ1,...,θ k =0 |R k (θ 0 , θ 1 , . . . , θ k ) | θ 1 W 1,∞ . . . θ k W 1,∞ C (n, k, f W k+1,1 ) θ 0 W 1,∞ k i1,...,i k =1 k m=0 Im⊆ 1,k cardIm=m q∈S Im |R(θ 0 )|.
We emphasize the fact that even if the notation omitted it, the R(θ 0 ) in (47)(48) depends on n, k, and f , but also on i 1 , . . . , i k , m, I m , and q. Since all the sums are nite, we can take the maximum of these R(θ 0 ) for example, and we end up with ,k,f,θ) depending only on n, k, f and θ 0 . We have thus established that the map ( 44) is dierentiable at the origin i.e.

|||R k (θ 0 , •, . . . , •)||| θ 0 W 1,∞ R(n, k, f, θ 0 ), where |R(n, k, f, θ 0 )| → 0 as θ 0 W 1,∞ → 0 with R(n
F Ω : θ ∈ W 1,∞ → F [(I +θ)(Ω)] is k + 1 times dierentiable at the origin for any measurable subset Ω of R n .
We now show that F Ω is k + 1 times dierentiable at any point of B 0,1 ∩ W 1,∞ . Let θ 0 ∈ W 1,∞ be such that θ 0 C 0,1 < 1. From Proposition 4.1, the map I +θ 0 has a Lipschitz continuous inverse. In particular, we deduce that Ω 0 := (I + θ 0 )(Ω) is measurable. Consequently, from the foregoing, the function

F Ω0 : θ ∈ W 1,∞ → F [(I + θ)(Ω 0 )] is k + 1 times dierentiable at the origin. Let ε > 0 and we set := ε(1 -θ 0 C 0,1 ) k+1 > 0. There exists δ ∈]0, 1[ such that for any θ ∈ W 1,∞ such that θ W 1,∞ < δ, we have: |||D k θ F Ω0 -D k 0 F Ω0 -D k+1 0 F Ω0 (•, . . . , •, θ)||| θ W 1,∞ .
Proceeding as in [START_REF] Szabo | Modern Quantum Chemistry: introduction to advanced electronic structure theory[END_REF], we observe that for any

h ∈ W 1,∞ such that h W 1,∞ < δ(1 -θ 0 C 0,1 ), we have θ 0 + h C 0,1 < 1 so we can write for any (θ 1 , . . . , θ k ) ∈ W 1,∞ × . . . × W 1,∞ : D k θ0+h F Ω -D k θ0 F Ω (θ 1 , . . . , θ k ) -D k+1 0 F Ω0 θ 1 • (I + θ 0 ) -1 , . . . , θ k • (I + θ 0 ) -1 , h • (I + θ 0 ) -1 = D k θ F Ω0 -D k 0 F Ω0 θ 1 • (I + θ 0 ) -1 , . . . , θ k • (I + θ 0 ) -1 -D k+1 0 F Ω0 θ 1 • (I + θ 0 ) -1 , . . . , θ k • (I + θ 0 ) -1 , θ ,
where we have set θ := h

• (I + θ 0 ) -1 . As in (42), we have θ W 1,∞ h W 1,∞
1-θ0 C 0,1 < δ so we get:

|[D k θ0+h F Ω -D k θ0 F Ω ](θ 1 , . . . , θ k ) -D k+1 0 F Ω0 [θ 1 • (I + θ 0 ) -1 , . . . , θ k • (I + θ 0 ) -1 , h • (I + θ 0 ) -1 ]| |||D k θ F Ω0 -D k 0 F Ω0 -D k+1 0 F Ω0 (•, . . . , •, θ)||| k l=1 θ l • (I + θ 0 ) -1 W 1,∞ h • (I + θ 0 ) -1 W 1,∞ k l=1 θ l W 1,∞ 1 -θ 0 C 0,1 (1 -θ 0 C 0,1 ) k+1 =ε h W 1,∞ k l=1 θ l W 1,∞ .

Consequently, we obtain for any

h ∈ W 1,∞ such that h W 1,∞ < δ(1 -θ C 0,1 ): |||D k θ0+h F Ω -D k θ0 F Ω -D k+1 0 F Ω0 [(•) • (I + θ 0 ) -1 , . . . , (•) • (I + θ 0 ) -1 , h • (I + θ 0 ) -1 ]||| ε h W 1,∞ . Since (θ 1 , . . . , θ k , h) ∈ (W 1,∞ ) k+1 → D k+1 0 F Ω0 [θ 1 • (I + θ 0 ) -1 , . . . , θ k • (I + θ 0 ) -1 , h • (I + θ 0 ) -1
] is a continuous symmetric (k + 1)-linear form, we have proved that 1 and its dierential is well dened by (44) with k + 1 instead of k.

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is k + 1 times dierentiable at any point of W 1,∞ ∩ B 0,
Then, we now show that the (k + 1)-th order dierential of F Ω is continuous at the origin. Let θ 0 ∈ W 1,∞ be such that θ 0 W 1,∞ < δ, where we recall that δ > 0 is such that the Jacobian determinant of I + θ 0 is positive. Since we have just proved that (44) holds true for k + 1, we can rigorously use the same arguments than we used in the beginning of the proof in order to get that (45) holds true for k + 1 instead of k. Moreover, considering the expressions ( 43) and (45) with k + 1 instead of k, in each product/sum concerning j ∈ I l , we make a change of indices u := p -1 I l (j) so as to be able to order the product/sum from u = 1 to u = l. Using again the previous notation we introduced below (46), we thus have for any (θ 1 , . . . , θ k+1 ) ∈ W 1,∞ × . . . × W 1,∞ :

D k+1 θ0 F Ω -D k+1 0 F Ω (θ 1 , . . . , θ k+1 ) = n i1,...,i k+1 =1 k+1 l=0 I l ⊆ 1,k+1 card I l =l q∈S I l s(q) Ω j∈ 1,k+1 j / ∈I l [θ j ] ij l+1 u=0 a u - l+1 u=0 b u = n i1,...,i k+1 =1 k+1 l=0 I l ⊆ 1,k+1 card I l =l q∈S I l s(q) Ω j∈ 1,k+1 j / ∈I l [θ j ] ij l+1 u0=0 u0-1 u=0 b u (a u0 -b u0 ) l+1 u=u0+1 a u ,
where (a j , b j ) 1 j l+1 are dened as before (see below (46) where m has been replaced by l), but where k is replaced by k + 1 in the denition of (a 0 , b 0 ). Therefore, we can now estimate as before each term in the last equality in order to get

|||D k+1 θ0 F Ω -D k+1 0 F Ω ||| → 0 as θ 0 W 1,∞ → 0.
Let us detail this procedure. First, we can apply Proposition 4.6 to the maps ∂ k+1-l f ∈ L 1 , l ∈ 0, k + 1 , use the Cauchy-Schwarz inequality with (61), and combine relations (62)(63) with θ 0 W 1,∞ < δ < 1. We deduce that:

(a 0 -b 0 ) l u=1 a u a l+1 L 1 (n -1)! n 1 -δ l+1 R(θ 0 ) j∈I l θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 W 1,∞ → 0. Hence, we have estimated the rst term of the rst sum. We proceed similarly for the other ones. Using the L 1 -norm for the maps ∂ k+1-l • f , l ∈ 0, k + 1 , and the L ∞ -norm for the remaining terms, we get from the Cauchy-Schwarz inequality with (61), relations (62)(63) with θ 0 W 1,∞ < δ < 1, and Proposition 4.3:

l u0=1 b 0 u0-1 u=1 b u (a u0 -b u0 ) l u=u0+1 a u a l+1 L 1 l √ n(n -1)! n 1 -δ l f W k+1,1 R(θ 0 ) j∈I l θ j C 0,1 ,
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. The same arguments combined with Proposition 4.4 lead to:

b 0 l u=1 b u (a l+1 -b l+1 ) L 1 f W k+1,1   j∈I l θ j C 0,1   R(θ 0 ),
where |R(θ 0 )| → 0 as θ 0 C 0,1 → 0. Gathering the three last estimations and observing that l k + 1 with δ ∈]0, 1[ only depending on n, we obtain:

l+1 u0=0 u0-1 u=0 b u (a u0 -b u0 ) l+1 u=u0+1 a u L 1 C (n, k, f W k+1,1 )   j∈I l θ j W 1,∞   R(θ 0 ),
where C(n, k, f W k+1,1 ) > 0 is a xed constant depending only on n, k, and f W k+1,1 , and where |R(θ 0 )| → 0 as θ 0 W 1,∞ → 0. We use this last inequality in order to estimate the last expression obtained for D k+1 θ0 F Ω -D k+1 0 F Ω . We deduce that:

|||D k+1 θ0 F Ω -D k+1 0 F Ω ||| := sup (θ1,...θ k+1 )∈(W 1,∞ ) k+1 θ1,...,θ k+1 =0 | D k+1 θ0 F Ω -D k+1 0 F Ω (θ 1 , . . . , θ k+1 ) | θ 1 W 1,∞ . . . θ k+1 W 1,∞ C (n, k, f W k+1,1 ) n i1,...,i k+1 =1 k+1 l=0 I l ⊆ 1,k+1 card I l =l q∈S I l R(θ 0 ).
As before, even if the notation omitted it, the R(θ 0 ) in the previous estimations were depending on n, k, and f , but also on i 1 , . . . , i k , l, I l , and q. Since all the sums are nite, we can take the maximum of these R(θ 0 ) and we end up with

|||D k+1 θ0 F Ω -D k+1 0 F Ω ||| R(n, k, f, θ 0 ), where |R(n, k, f, θ 0 )| → 0 as θ 0 W 1,∞ → 0 with R(n,
k, f, θ) depending only on n, k, f and θ 0 . Therefore, we have established that the map D k+1

• F Ω : θ 0 ∈ W 1,∞ → D k+1 θ0 F Ω ∈ L k+1 c ((W 1,∞ ) k+1 , R) is continuous at the origin i.e. F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is k + 1 times continuously dierentiable at the origin for any measurable subset Ω of R n .
Finally, it remains to establish that the map D k+1

• F Ω is continuous at any point of B 0,1 ∩ W 1,∞ .
The arguments are the same than those used to obtain the (k + 1)-th order dierentiability at any point of W 1,∞ ∩ B 0,1 from the one at the origin. Let θ 0 ∈ W 1,∞ be such that θ 0 C 0,1 < 1. From Proposition 4.1, the map I + θ 0 has a Lipschitz continuous inverse. In particular, we deduce that Ω 0 := (I + θ 0 )(Ω) is measurable. From the foregoing, the map

F Ω0 : θ ∈ W 1,∞ → F [(I + θ)(Ω 0 )] is k + 1 times continuously dierentiable at the origin. Let ε > 0 and set := ε(1 -θ 0 C 0,1 ) k+1 > 0. There exists δ ∈]0, 1[ such that for any θ ∈ W 1,∞ such that θ W 1,∞ < δ, we have the inequality |||D k+1 θ F Ω0 -D k+1 0 F Ω0 |||
. Proceeding as in [START_REF] Szabo | Modern Quantum Chemistry: introduction to advanced electronic structure theory[END_REF], we observe that for any h ∈ W 1,∞ such that h W 1,∞ < δ(1 -θ 0 C 0,1 ), we have the estimation θ 0 + h C 0,1 < 1 so we can write for any

(θ 1 , . . . , θ k+1 ) ∈ W 1,∞ × . . . × W 1,∞ : D k+1 θ0+h F Ω -D k+1 θ0 F Ω (θ 1 , ..., θ k+1 ) = D k+1 θ F Ω0 -D k+1 0 F Ω0 [θ 1 •(I + θ 0 ) -1 , ..., θ k+1 •(I + θ 0 ) -1 ],
where we have set θ := h

• (I + θ 0 ) -1 . As in (42), we have θ W 1,∞ h W 1,∞
1-θ0 C 0,1 < δ so we get:

|[D k+1 θ0+h F Ω -D k+1 θ0 F Ω ] (θ 1 , . . . , θ k+1 ) | |||D k+1 θ F Ω0 -D k+1 0 F Ω0 ||| k+1 l=1 θ l • (I + θ 0 ) -1 W 1,∞ (1 -θ 0 C 0,1 ) k+1 k+1 l=1 θ l W 1,∞ = ε k+1 l=1 θ l W 1,∞ .
Hence, we obtain 1 . Consequently, we have proved that the statement is true for k = 0, and that if it is true for an integer k 0, then it is true for k + 1 provided f ∈ W k+1,1 . Therefore, by induction, for any integer k 0 1, if f ∈ W k0,1 , then we obtain recursively that for any k ∈ 1, k 0 , the functional (40) is k times shape dierentiable at any measurable subset of R n , its k-th-order shape derivative being well dened by (43). Moreover, 1 , its k-th-order dierential map being well dened by (44), which concludes the proof of Theorem 3.2.

|||D k+1 θ0+h F Ω -D k+1 θ0 F Ω ||| ε for any h ∈ W 1,∞ such that h W 1,∞ < δ(1-θ C 0,1 ) i.e. D k+1 • F Ω is continuous at any point W 1,∞ ∩ B 0,
F Ω : θ ∈ W 1,∞ → F [(I +θ 0 )(Ω)] is k times continuously dierentiable on W 1,∞ ∩B 0,

The intermediate case of Lipschitz regularity

In this section, we show that further regularity on the boundary and the vector elds yields to express (43) into a divergence form. Applying the Trace Theorem [17, Section 4.3], we obtain a new relation for the shapes derivatives of a volume integral. Moreover, if we assume that one of the vector elds is normal to the boundary, then the expression can be signicantly simplied.

Theorem 3.3. Let n 2, k 0 1, and f ∈ W k0,1 (R n , R).
We consider an open bounded subset Ω of R n with a Lipschitz boundary. Then, the map

F Ω : θ ∈ W 1,∞ ∩ C 1 → F [(I + θ)(Ω)] is k 0 times
Fréchet dierentiable at the origin and for any k ∈ 1, k 0 , its dierential of order k at the origin is given for any

(θ 1 , . . . , θ k ) ∈ (W 1,∞ ∩ C 1 ) × . . . × (W 1,∞ ∩ C 1 ) by: D k 0 F Ω (θ 1 , . . . , θ k ) = ∂Ω   n i1,...,i k-1 =1 ∂ k-1 f k-1 j=1 ∂x ij k-1 j=1 [θ j ] ij   (θ k ) n dA + n i1,...,i k-1 =1 k-1 l=1 I l ⊆ 1,k-1 card I l =l p∈S I l s (p) ∂Ω ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [θ j ] ij     (θ k ) n j∈I l [D • θ j ] ij i p(j) - j1∈I l [D • θ j1 (θ k )] ij 1 [n Ω ] i p(j 1 ) j∈I l j =j1 [D • θ j ] ij i p(j)     dA. (49)
Moreover, in the case where θ k is normal to the boundary i.e. if we have θ k (x) = (θ k ) n (x)n Ω (x) for any point x ∈ ∂Ω, then relation (49) takes the following form:

D k 0 F Ω (θ 1 , . . . , θ k ) = n i1,...,i k-1 =1 k-1 l=0 I l ⊆ 1,k-1 card I l =l p∈S I l s(p) ∂Ω ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [θ j ] ij j∈I l [D ∂Ω θ j ] ij i p(j) (θ k ) n dA.
(50) Finally, the map 1 and for any k ∈ 1, k 0 , its k-th-order dierential is well dened by the following continuous map:

F Ω : θ ∈ W 1,∞ ∩ C 1 → F [(I + θ)(Ω)] is k 0 times continuously dierentiable at any point of W 1,∞ ∩ C 1 ∩ B 0,
D k • F Ω : W 1,∞ ∩ C 1 ∩ B 0,1 -→ L k c W 1,∞ ∩ C 1 k , R θ 0 -→ (θ 1 , ..., θ k ) → D k 0 F (I+θ0)(Ω) [θ 1 • (I + θ 0 ) -1 , ..., θ k • (I + θ 0 ) -1 ] (51) where D k 0 F (I+θ0)(Ω)
is the k-th-order shape derivative of ( 40) at (I +θ 0 )(Ω) given by (49) in general and by (50) if θ k is normal to the boundary. Remark 3.4. We emphasize here the fact that even if the right member of (49) is not symmetric with respect to the vector elds, the shape derivative (43) is a continuous symmetric k-linear form. In fact, the symmetry of a derivative is a consequence of the Fréchet dierentiability. Hence, (49) is a symmetric k-linear form and (50) also holds true if any of the vector elds is normal to ∂Ω.

Proof. Let n 2, k 0 1, f ∈ W k0,1 (R n , R),
and consider an open bounded set Ω ⊂ R n with a Lipschitz boundary. Since W 1,∞ ∩ C 1 is equipped with the W 1,∞ -norm, we can apply Theorem 3.2 to the restriction map

F Ω : θ ∈ W 1,∞ ∩ C 1 → F [(I + θ)(Ω)], which is thus k 0 times continuously dierentiable on W 1,∞ ∩ C 1 ∩ B 0,1 . For any k ∈ 1, k 0 , its k-th-order dierential is well dened by (51), where D k 0 F (I+θ0)(Ω)
is the k-th-order shape derivative of ( 40) at (I + θ 0 )(Ω) given by (43) for the moment. We now aim to use the additional regularity we made on the boundary and the vector elds in order to improve (43). First, we assume that (θ 1 , . . . , θ k ) ∈ W 2,∞ × . . . × W 2,∞ . The proof consists in establishing that in this case, the right member of (43) can be expressed in the following divergence form:

n i1,...,i k =1 k-1 l=0 I l ⊆ 1,k-1 card I l =l p∈S I l s (p) Ω ∂ ∂x i k      ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [θ j ] ij j∈I l [D • θ j ] ij i p(j) [θ k ] i k      - n i1,...,i k =1 k-1 l=1 I l ⊆ 1,k-1 card I l =l j1∈I l p∈S I l s (p) Ω ∂ ∂x i p(j 1 )      ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [θ j ] ij [θ k ] i k [D • θ j1 ] ij 1 i k j∈I l j =j1 [D • θ j ] ij i p(j)     .
(52) We emphasize the fact that (52) is equal to the right member of (43) only for W 2,∞ -vector elds. Note also that if this last assertion is true, then we obtain that (49) holds true by applying the Trace Theorem [17, Section 4.3] to (52). More precisely, we obtain that (49) holds true for W 2,∞ -vector elds and we extend the result to the W 1,∞ ∩ C 1 -ones from standard approximating arguments. Indeed, for any θ ∈ W 1,∞ ∩ C 1 , there exists a sequence (θ k ) k∈N of elements in W 2,∞ such that θ k and [D • θ k ] ij respectively converges to θ and [D • θ] ij uniformly on any compact subset of R n as k → +∞ and for any (i, j) ∈ 1, n 2 (consider the usual mollier [17, Section 4.2.1 Theorem 1]). Therefore, the main diculty here is to check by direct calculations that (52) is equal to the right member of (43). Let us now detail the great lines of this (tedious) calculation. On the one hand, we expand the i k -partial derivative in the rst integral of (52), which is composed of a product of four terms. This expansion (from left to right) thus yields to the sum of four terms respectively denoted by A 1 , A 2 , A 3 and A 4 . Similarly, the i p(j1) -partial derivative in the second integral of (52) is expanded and yields to the sum of ve terms referred to as B 1 , B 2 , B 3 , B 4 , and B 5 . Note that the terms A 2 , A 3 , B 2 and B 5 take a partial derivative with respect to a product (on j / ∈ I l or j ∈ I l ). Hence, a new sum appears for the expansion of these terms and the notation j 0 refers to it. On the other hand, the right member of (43) is divided into three situations as in the proof of Theorem 3.2 (see below (45)). We denote by C 1 the case where k / ∈ I l , by C 2 the case where k ∈ I l and p(k) = k, and by C 3 the case where k ∈ I l and p(k) = k. With these notation in mind, we get from the foregoing that (49) holds true if we can prove that:

A 1 + A 2 + A 3 + A 4 + B 1 + B 2 + B 3 + B 4 + B 5 = C 1 + C 2 + C 3 .
More precisely, we are going to check that

C 1 = A 1 , C 2 = A 4 , C 3 = B 3 , A 2 +B 2 = -B 1 , A 3 = -B 4 ,
and B 5 = 0. Since the term C 1 corresponds to the situation where k / ∈ I l , the sum on l can stop at k -1 and the subset I l is chosen in 1, k -1 , from which we immediately get C 1 = A 1 . Concerning the relation involving C 2 , we are in the situation where k ∈ I l and p(k) = k. The sum on l can thus start at one and it is equivalent to search for I l-1 ⊆ 1, k -1 and q ∈ S I l-1 by setting I l := I l-1 ∪ {k} and p := q on I l-1 with p(k) := k. Note also that s(q) = s(p). Re-indexing the summation on l by setting m := l -1, we deduce that C 2 = A 4 . Then, C 3 corresponds to the case where k ∈ I l and p(k) = k so the sum on l can start at two and we can search for I l-1 ⊆ 1, k -1 by setting I l := I l-1 ∪ {k}. We can also partition the sum on p ∈ S I l such that p(k) = k by xing the element k. In other words, we get a sum on j 0 ∈ I l-1 followed by a sum on p ∈ S I l such that p(j 0 ) = k. We can re-index this last sum by setting q := p • t, where t only exchanges j 0 and k. We are back to a summation on q ∈ S I l with q(k) = k i.e. to the situation of C 2 but in this case we have s(q) = s(p • t) = s(p)s(t) = -s(p). Proceeding as for C 2 , we deduce that C 3 = B 3 . Then, we decompose the term B 1 into two disjoint situations. On the one hand, we impose p(j 1 ) = j 1 , which is equivalent to choose I l-1 ⊆ 1, k -1 and j 1 ∈ 1, k -1 \I l-1 then set I l := I l-1 ∪ {j 1 }. Similarly, the sum on p ∈ S I l with p(j 1 ) = j 1 is reduced to a sum on q ∈ S I l-1 by setting p := q on I l-1 and p(j 1 ) := j 1 . Note also that s(p) = s(q). Re-indexing the summation on l by setting m := l -1, we get that this expression yields to -A 2 . On the other hand, we have p(j 1 ) = j 1 and we can partition this sum by xing the element p(j 1 ). More precisely, searching for I l ⊆ 1, k -1 is equivalent to search for I l-1 ⊆ 1, k -1 and j 0 ∈ 1, k -1 \I l-1 by setting I l := I l-1 ∪ {j 0 }. Similarly, the sum on j 1 ∈ I l followed by the one p ∈ S I l such that p(j 1 ) = j 1 is replaced by a sum on j 1 ∈ I l-1 followed by one on p ∈ S I l with p(j 1 ) = j 0 . We can next re-arrange the summation of the permutations by setting q := p • t, where t is only exchanging j 0 and j 1 . We are thus back in the previous situation where q(j 1 ) = j 1 but a negative sign now appears since s(q) = s(p • t) = s(p)s(t) = -s(p). Proceeding as before, we get that this term is equal to -B 2 . Hence, we have proved that B 1 = -A 2 -B 2 . Comparing the two terms A 3 and B 4 , we immediately get that A 3 = -B 4 by observing that the sum on l in A 3 can start at one since I l is not empty in this case. Finally, it remains to check that B 5 = 0. This is the term with A 3 and B 4 which needs the W 2,∞ -regularity assumption on the vector elds. Performing a change of variables the permutations by setting q := p • t where t exchanges the two dierent indices appearing in the second-order partial derivatives of f , one can notice that we obtain the same expression, up to a sign since s(q) = s(p • t) = s(p)s(t) = -s(p). We deduce that B 5 = -B 5 i.e. B 5 = 0. Therefore, we have proved that (49) holds true. Moreover, for any

θ 0 ∈ W 1,∞ ∩ C 1 ∩ B 0,1 , the domain (I + θ 0 )(Ω) also has a Lipschitz boundary ∂[(I + θ 0 )(Ω)] = (I + θ 0 )(∂Ω) and θ • (I + θ 0 ) -1 ∈ W 1,∞ ∩ C 1 for any θ ∈ W 1,∞ ∩ C 1 .
Hence, we deduce that we can use (49) instead of (43) to dene D k 0 F (I+θ0)(Ω) in (51). It remains to study the case where θ k is normal to the boundary. Again, the calculations are tedious so we only sketch the proof. We assume θ k = (θ k ) n n Ω on ∂Ω and we deduce from (49):

D k 0 F Ω (θ 1 , . . . , θ k ) = ∂Ω   n i1,...,i k-1 =1 ∂ k-1 f k-1 j=1 ∂x ij k-1 j=1 [θ j ] ij   (θ k ) n dA + n i1,...,i k-1 =1 k-1 l=1 I l ⊆ 1,k-1 card I l =l p∈S I l s (p) ∂Ω ∂ k-1-l f j∈ 1,k-1 j / ∈I l ∂x ij j∈ 1,k-1 j / ∈I l [θ j ] ij (θ k ) n     j∈I l [D • θ j ] ij i p(j) - j1∈I l [D • θ j1 (n Ω )] ij 1 [n Ω ] i p(j 1 ) j∈I l j =j1 [D • θ j ] ij i p(j)     dA. (53)
Let us now distinguish the tangential and normal part of the dierential operator. Therefore, we expand the product j ∈ I l in (53) as follows:

j∈I l [D • θ j ] ij i p(j) = j∈I l [D ∂Ω θ j ] ij i p(j) + [D • θ j (n Ω )] ij [n Ω ] i p(j) = l m=0 Jm⊆I l card Jm=m   j∈Jm [D • θ j (n Ω )] ij [n Ω ] i p(j)     j∈Im\Jm [D ∂Ω θ j ] ij i p(j)   .
First, note that the rst boundary integral in (53) corresponds to the case l = 0 in (50). Hence, we have to check that the remaining part of (50) is equal to the second integral in (53). This latter is the dierence of two terms denoted by A 1 and A 2 . Then, we expand the product j ∈ I m in A 1 as above while the same is done for A 2 in the product j ∈ I m \{j 1 }. The idea now consists in setting J m+1 := J m ∪ {j 1 } in A 2 . In particular, the summation on j 1 ∈ I l followed by the one on J m ⊆ I l \{j 1 } is equivalent to search for J m+1 ⊆ I l . Re-indexing the summation on m by setting m = m + 1, we deduce that A 2 is equal to A 1 , apart from the case m = 0 which is exactly the the remaining part of (50) we were talking about. Consequently, we have established that (50) holds true, concluding the proof of Theorem 3.3. Corollary 3.5. Consider the assumptions of Theorem 3.3 in the case k 0 = k = 1. Then, [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] is shape dierentiable at Ω and its shape derivative is given by the following continuous linear form:

∀θ ∈ W 1,∞ , D 0 F Ω (θ) = ∂Ω f θ n dA. (54)
Moreover, the map

F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is continuously dierentiable on W 1,∞ ∩ B 0,1
and its dierential is given by the following continuous map:

D • F Ω : W 1,∞ ∩ B 0,1 -→ L c W 1,∞ , R θ 0 -→ D θ0 F Ω := θ → D 0 F (I+θ0)(Ω) θ • (I + θ 0 ) -1 , (55)
where D 0 F (I+θ0)(Ω) is the shape derivative of [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] at (I + θ 0 )(Ω) given by (54).

Proof. Applying Theorem 3.2, we get that F Ω : θ ∈ W 1,∞ → F [(I + θ)(Ω)] is continuously dierentiable on W 1,∞ ∩ B 0,1 and its dierential is well dened by (55) but the shape derivative of ( 40) is given by:

∀θ ∈ W 1,∞ , D 0 F Ω (θ) = Ω div (f θ) .
Applying the Trace Theorem [17, Section 4.3], we get that (54) holds true for (W 1,∞ ∩ C 1 )-vector elds. We can extend the result to any θ ∈ W 1,∞ from standard approximating arguments. Indeed, for any θ ∈ W 1,∞ , there exists a sequence

(θ i ) i∈N ⊂ W 1,∞ ∩ C 1 converging to θ L ∞ -strongly, W 1

The specic case of C 1,1 -domains

In this section, we show that the C 1,1 -regularity of the boundary is enough to ensure the notion of partial derivative with respect to the domain at any order higher than two. We emphasize here a technical issue related to the case where the perturbations are normal to the boundary. The results of Theorem 3.6 that follows could have been found by inserting the relation θ j = (θ j ) n n Ω in (49). However, in order to do so, we have to dene D • n Ω whereas n Ω is a priori only dened on the boundary. This can be done be considering an extension N Ω ∈ W 1,∞ ∩ C 1 of the normal vector but this is possible only if ∂Ω is a C 2 -surface. Therefore, the great advantage of (50), apart from its simplicity, consists in expressing the shape derivatives with the tangential operator D ∂Ω , which an intrinsic notion. In particular, D ∂Ω n Ω can be dened via the local parametrization of the surface, for which we only need C 1,1 -regularity. This technical detail can be important in the applications since the C 1,1 -regularity has various geometrical characterizations (positive reach [START_REF] Federer | Curvature measures[END_REF], uniform ball property [START_REF] Dalphin | Some characterizations of a uniform ball property[END_REF], oriented distance function [START_REF] Delfour | Shapes and geometries: metrics, analysis, dierential calculus, and optimization[END_REF]Chapter 7]).

Annexes

In this section, we aim to derive all the technical material that was needed throughout the article.

The results presented here are standard [START_REF] Delfour | Shapes and geometries: metrics, analysis, dierential calculus, and optimization[END_REF]Chapter 9] [22, Chapter 5] [40, Chapter 2] and they are organized as follows. First, we recall some terminology about dierentiability in Banach spaces and we introduce the Sobolev norms in which we are interested. Then, we give some dierentiability results related to the inverse, the Jacobian determinant, and the composition operator.

Some denitions and notation

Let n 2 be an integer henceforth set. The space R n is equipped with its usual Euclidean structure: for any

(x, y) ∈ R n × R n , we have set x | y := n k=1 x k y k and |x| := x | x = n k=1 |x k | 2 .
More generally, any set E here refers to a real vector space provided with a norm • E . The set L c (E, F ) of continuous linear maps between two such spaces is endowed with its operator norm:

∀u ∈ L c (E, F ) , |||u||| := sup x∈E x =0 E u (x) F x E .
This norm is complete as soon as • F is complete, and L c (E, E) is a unitary Banach algebra [START_REF] Rudin | Real and complex analysis[END_REF]Chapter 18]. We also recall that if E is nite dimensional, then the norms dened on E are equivalent and complete [4, I 2 Section 3]. In this case, E is necessarily a Banach space and any linear map u : E → F is continuous. Moreover, a well-dened map g : E → F is said to be Fréchet dierentiable at a point x ∈ E if there exists a continuous linear map L x ∈ L c (E, F ) such that:

∀h ∈ E, g (x + h) = g (x) + L x (h) + h E R (h) ,
where R(h) F → 0 as h E → 0. In this case, the operator L x is unique, denoted by D x g, and called the dierential of g at the point x. If in addition, the map

D • g : y ∈ E → D y g ∈ L c (E, F )
is well dened around x and continuous at x, then we say that g is of class C 1 at x or continuously dierentiable at x. Similarly, we can proceed recursively for any integer k 2. Hence, if the map

D k-1 • g : y ∈ E → D k-1 y g ∈ L k-1 c (E k-1 , F
) is well dened around x and dierentiable at x, then we say that g is k times (Fréchet) dierentiable at x, and the dierential of D k-1 • g at x is identied with a continuous k-linear map, denoted by D k

x g and called the k-th-order dierential of g at x, via the following bijective linear isometry:

L c E, L k-1 c E k-1 , F -→ L k c E k , F y 0 → [
u y0 : (y 1 , ..., y k-1 ) → u y0 (y 1 , ..., y k-1 )] -→ (y 0 , y 1 , ..., y k-1 ) → u y0 (y 1 , ..., y k-1 ) , where L k c (E k , F ) is the set of continuous k-linear maps equipped with the norm:

∀u ∈ L k c E k , F , |||u||| := sup (x1,...,x k )∈E k (x1,...,x k ) =(0 E ,...0 E ) u (x 1 , . . . , x k ) F x 1 E . . . x k E . If in addition, the map D k • g : y ∈ E → D k y ∈ L k c (E k , F
) is well dened around x and continuous at the point x, then we say that g is of class C k at x or k times continuous dierentiable at x. Then, for any real p 1, we denote by L p the space of measurable maps from R n into R n whose p-th power is integrable, and by L ∞ the space of measurable maps from R n into R n that are essentially bounded. They are respectively endowed with their usual norm:

∀ (f , g) ∈ L p × L ∞ , f p := R n |f (x) | p dx 1 p and g ∞ := ess sup x∈R n |g (x) |,
where the integration is done with respect to the usual n-dimensional Lebesgue measure. We recall that each L p and L ∞ are Banach spaces [START_REF] Rudin | Real and complex analysis[END_REF]3.11 Theorem]. Moreover, for any measurable map f : R n → R which is locally integrable, we say that f is weakly dierentiable if there exists a measurable map g : R n → R n which is locally integrable, and such that:

∀ϕ ∈ C ∞ c , R n g (x) | ϕ (x) dx = - R n f (x) div ϕ (x) dx,
where C ∞ c refers to the set of smooth maps from R n into R n with compact support. In this case, the function g is unique, denoted by ∇f , and called the weak gradient of f . For any real p 1, we can now introduce the Sobolev space W 1,p as the set of functions f ∈ L p (R n , R) that are weakly dierentiable and whose weak gradients ∇f are functions of L p . Moreover, any W 1,p is a Banach space [5, Chapter 9] endowed with the norm:

∀f ∈ W 1,p , f 1,p := R n |f (x) | p dx 1 p + ∇f p .
The space C 0,1 of Lipschitz continuous maps from R n into R n is equipped with the norm:

∀θ ∈ C 0,1 , θ 0,1 := sup (x,x)∈R n ×R n x =x |θ (x) -θ (x) | |x -x| .
We recall that C 0,1 is not a Banach space i.e. the norm • 0,1 is not complete. We denote by B 0,1 := {θ ∈ C 0,1 , θ 0,1 < 1} the open unit ball of C 0,1 centred at the origin i.e. the set of Lipschitz contractions. We recall that C 0,1 can be identied with the subspace of continuous maps from R n into R n whose weak partial derivatives are functions of L ∞ [17, Section 4.2.3]. Moreover, any Lipschitz continuous map is dierentiable almost everywhere [17, Section 3.1.2] and we have θ 0,1 = ess sup x∈R n |||D x θ||| for any θ ∈ C 0,1 . We also introduce the space W 1,∞ = L ∞ ∩ C 0,1 of Lipschitz continuous bounded maps from R n into R n , provided with the norm:

∀θ ∈ W 1,∞ , θ 1,∞ := θ ∞ + θ 0,1 .
In particular, W 1,∞ is a Banach space [5, Proposition 9.1] and I : x ∈ R n → x ∈ R n denotes the identity map. Finally, we can dene recursively for any integer k 2 the Sobolev spaces W k,p as the set of all maps f : R n → R that are in W 1,p and such that each component of its weak gradient is a function of W k-1,p . It can be endowed with the norm:

∀f ∈ W k,p , f k,p := f 1,p + n i=1 ∂ i f k-1,p .
Similarly, the space W k,∞ is dened recursively as the set of maps θ : R n → R n that are in W 1,∞ such that their partial derivatives are functions of W k-1,∞ . It is equipped with the norm:

∀θ ∈ W k,∞ , θ k,∞ := θ 1,∞ + n i=1 ∂ i θ k-1,∞ .
To conclude, W k,p and W k,∞ are Banach spaces for any integer k 2 [5, above Section 9.2] and we use the specic notation H k := W k,2 because it is an Hilbert space [5, Proposition 9.1].

4.2 About the dierentiability related to the inverse operator Proposition 4.1. Let θ ∈ B 0,1 . Then, I + θ is a (1 + θ 0,1 )-Lipschitz continuous map which is invertible, and its inverse (I + θ) -1 is a 1 1-θ 0,1 -Lipschitz continuous map satisfying:

(I + θ) -1 -I 0,1 θ 0,1 1 -θ 0,1 . (58)
In particular, the map θ ∈ C 0,1 → (I + θ) -1 ∈ C 0,1 is well dened on B 0,1 and it is continuous at the origin. If in addition, we assume that θ is bounded, then we have (I + θ) -1 -I ∈ W 1,∞ and the following estimations hold true:

(I + θ) -1 -I 1,∞ θ 1,∞ 1 -θ 0,1 , (59)
(I + θ) -1 -I + θ ∞ θ 1,∞ 2 2 .
(60)

In particular, the map θ ∈ W 1,∞ → (I + θ) -1 -I ∈ W 1,∞ is well dened on W 1,∞ ∩ B 0,1 , and it is continuous at the origin. Moreover, the map θ ∈ W 1,∞ → (I + θ) -1 -I ∈ L ∞ is dierentiable at the origin, its dierential being the opposite of the inclusion map from W 1,∞ into L ∞ .

Proof. Let θ ∈ B 0,1 . First, from the triangle inequality, we get the (1 + θ 0,1 )-Lipschitz continuity of I + θ. Then, for any z ∈ R n , the map x ∈ R n → z -θ(x) ∈ R n is a contraction thus the Banach Fixed-Point Theorem [5, Theorem 5.7] asserts there exists a unique point x z ∈ R n such that z -θ(x z ) = x z i.e. I + θ is a bijective map. Moreover, since (I + θ) -1 = I -θ • (I + θ) -1 , we obtain (I +θ) -1 0,1 1+ θ 0,1 (I +θ) -1 0,1 , from which we deduce that (I +θ) -1 is a 1 1-θ 0,1 -Lipschitz continuous map. Similarly, we have (I + θ) -1 -I 0,1 = -θ • (I + θ) -1 0,1 θ 0,1 (I + θ) -1 0,1 so relation (58) holds true. Finally, if we now assume that θ is bounded, then we also get:

               (I + θ) -1 -I 1,∞ θ ∞ + θ 0,1 1 -θ 0,1 = θ 1,∞ -θ 0,1 θ ∞ 1 -θ 0,1 θ 1,∞ 1 -θ 0,1 (I + θ) -1 -I + θ ∞ = θ -θ • (I + θ) -1 ∞ θ 0,1 I -(I + θ) -1 ∞ θ ∞ θ 2 1,∞ 4
.

To conclude the proof of Proposition 4.1, (I -θ) -1 -I ∈ W

∈ C 0,1 → D • θ ∈ L ∞ (R n , R n 2 )
is well dened, linear, and continuous. Moreover, for almost every point x ∈ R n , we have D x (I + θ) = I + D x θ and if we assume that θ 0,1 < 1, then we also have

D (I+θ)(x) [(I + θ) -1 ] = (I + D x θ) -1 for almost every point x ∈ R n .
Proof. Let θ ∈ C 0,1 . First, we dene Def (θ) as the set of points in R n for which θ is dierentiable and Rademacher's Theorem [17, Section 3.1.2] ensures that θ is dierentiable almost everywhere. Then, the dierential D x θ of θ at any x ∈ Def (θ) is a well-dened linear map, thus identied with its (n×n)-matrix representation in the canonic basis of R n denoted by (e 1 , . . . , e n ). Hence, the map

D • θ : x ∈ Def (θ) → D x θ ∈ R n 2 is measurable if and only if [D • θ] ij : x ∈ Def (θ) → ∂ j θ i (x) ∈ R is measurable for any (i, j) ∈ 1, n 2
, which is the case since it is respectively the pointwise limits of the continuous maps

(θ k ij ) k∈N : x ∈ R n → k[θ i (x+ 1 k e j )-θ i (x)] ∈ R.
Moreover, for any x ∈ Def (θ), we have:

D x θ R n 2 := n i,j=1 |∂ j θ i (x) | 2 √ n |D x θ (e j0 ) | √ n ess sup x∈R n |||D x θ||| = √ n θ 0,1 , (61)
where

j 0 ∈ 1, n satises |∂ j0 θ i (x)| = max 1 j n |∂ j θ i (x)|. Hence, θ ∈ C 0,1 → D • θ ∈ L ∞ (R n , R n 2 )
is a well-dened map, which is also linear (thus continuous by (61)). Indeed, for any λ ∈ R and any

(θ 1 , θ 2 ) ∈ C 0,1 × C 0,1 , we have D x θ 1 + λD x θ 2 = D x (θ 1 + λθ 2 ) for any x ∈ Def (θ 1 ) ∩ Def (θ 2 )
i.e. almost everywhere by Rademacher's Theorem. Hence, we get D

• (θ 1 + λθ 2 ) = D • θ 1 + λD • θ 2 . Similarly,
4.3. Let θ ∈ B 0,1 . Then, (I + D • θ) -1 : x ∈ R n → (I + D x θ) -1 ∈ R n 2 is well dened
almost everywhere and it is a measurable map satisfying for almost every point x ∈ R n :

(I + D x θ) -1 R n 2 √ n 1 -θ 0,1 . (62)
In particular, the map θ ∈ C 0,1 → (I + D

• θ) -1 ∈ L ∞ (R n , R n 2 )
is well dened on B 0,1 . Moreover, it is dierentiable at the origin and its dierential at the origin is given by the continuous linear map

θ ∈ C 0,1 → -D • θ ∈ L ∞ (R n , R n 2 ).
Proof. Let θ ∈ B 0,1 . First, from Lemma 4.2, we deduce that the map D

• θ : x ∈ R n → D x θ ∈ R n 2
is well dened for almost every point x ∈ R n and the matrix I + D x θ is invertible since we have |||D x θ||| θ 0,1 < 1 [33, 18.3]. Hence, the map (I + D • θ) -1 is well dened almost everywhere. Moreover, it is measurable as the composition between D • θ, which is measurable by Lemma 4.2, and the map A ∈ {B ∈ R n 2 , |||B||| < 1} → (I + A) -1 ∈ R n 2 which is continuous [33, 18.4]. Then, we use successively Lemma 4.2, relation (61) applied to (I + θ) -1 , and Proposition 4.1 in order to get for almost every point x ∈ R n :

(I + D x θ) -1 R n 2 D • [(I + θ) -1 ] L ∞ (R n ,R n 2 ) √ n (I + θ) -1 0,1 √ n 1 -θ 0,1 . Hence, θ ∈ C 0,1 → (I + D • θ) -1 ∈ L ∞ (R n , R n 2 )
is well dened on B 0,1 . It remains to prove that it is dierentiable at the origin. From Lemma 4.2, the map f :

θ ∈ C 0,1 → D • θ ∈ L ∞ (R n , R n 2 )
is well dened, linear and continuous. In particular, f is dierentiable at any point and its dierential at any point is the map f itself. In addition, the map g :

A ∈ {B ∈ R n 2 , |||B||| < 1} → (I +A) -1 ∈ R n 2
is dierentiable at the origin [33, 18.4] and its dierential is given by D 0 g :

A ∈ R n 2 → -A ∈ R n 2 .
We deduce that the map g

• f : θ ∈ C 0,1 → (I + D • θ) -1 ∈ L ∞ (R n , R n 2
) is dierentiable at the origin, and its dierential is given by the following continuous linear map:

In particular, the map θ ∈ C 0,1 → det[D • (I + θ)] ∈ L ∞ (R n , R) is well dened on B 0,1 . Moreover, it is dierentiable at the origin and its dierential is given by the divergence operator i.e. by the continuous linear map θ ∈ C 0,1 → div(θ) := trace(D • θ) ∈ L ∞ (R n , R).

Proof. Let θ ∈ B 0,1 . From Lemma 4.2, the map

D • θ : x ∈ R n → D x θ ∈ R
I p(i)i + ∂ i θ p(i) (x) (64)
Expanding the product and using the fact that I p(j)j = 1 if and only if j = p(j), we deduce that:

Then, note that ∂ i θ p(i) (x) is the p(i)-th component of the vector D x θ(e i ) where e i is the unit vector whose components are zero except the i-th one which is equal to one. Therefore, we have: (n -i) θ k 0,1 .

The last inequality comes from the fact that i∈I k θ 0,1 = θ k 0,1 does not depend on p and I k . We can thus remove it from the corresponding sums for which card{p ∈ S n | ∀j / ∈ I k , p(j) = j} = k! and card{I k ⊆ 1, n , card I k = k} = n k = n(n-1)...(n-k+1) k!

. Hence, we get: The last inequality holds true because θ 0,1 < 1 and the geometric series k∈N θ k 0,1 converges. Hence, estimation (63) holds true. Finally, the map J : θ ∈ C 0,1 → det[D • (I + θ)] ∈ L ∞ (R n , R) is the composition of the determinant det : R n 2 → R with the ane map I + f , where f is its linear part dened as f : θ ∈ C 0,1 → D • θ ∈ L ∞ (R n , R n 2 ). Since the determinant is dierentiable and since f is linear and continuous by Lemma 4.2, we deduce that J is dierentiable at the origin and its dierential is given by: To conclude, the divergence operator is the dierential of J = det • (I + f ) at the origin. 4.4 About the dierentiability related to the composition operator Proposition 4.6. Let f ∈ L 1 (R n , R). Then, the map θ ∈ C 0,1 → f • (I + θ) ∈ L 1 (R n , R) is well dened on B 0,1 . Moreover, θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is continuous at the origin. If in addition, we have f ∈ W 1,1 , then θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is dierentiable at the origin and its dierential is given by the continuous linear map θ ∈ W 1,∞ → ∇f | θ ∈ L 1 (R n , R).

Proof. Let f ∈ L 1 (R n , R). First, we check that the map θ ∈ C 0,1 → f • (I + θ) ∈ L 1 (R n , R) is well dened around the origin. Let θ ∈ B 0,1 . The function f • (I + θ) is measurable as the composition of the Lipschitz continuous map (I + θ) with the measurable map f . Proposition 4.1 ensures that the map (I + θ) has a Lipschitz continuous inverse, from which we deduce for almost every x ∈ R n : Consequently, using this last observation, relation (67), and the change of variables formula valid for any Lipschitz continuous map [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 3.3.3], we get:

Hence, we obtain f • (I + θ) ∈ L 1 (R n , R) for any f ∈ L 1 (R n , R) and any θ ∈ B 0,1 . Then, we prove that the map θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is continuous at the origin. Let θ ∈ W 1,∞ be such that θ 1,∞ 1 2 . We proceed by a density argument: there exists a sequence of smooth maps (f i ) i∈N : R n → R with compact support converging to f in L 1 (R n , R) [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF]Corollary 4.23]. On the one hand, we have θ 0,1 1 2 < 1 so the foregoing holds true. We can apply the arguments of (68) to the map f -f i in order to get for any i ∈ N:

f • (I + θ) -f i • (I + θ) L 1 (R n ,R) n! f -f i L 1 (R n ,R) (1 -θ 0,1 ) n n!2 n f -f i L 1 (R n ,R) .
On the other hand, since the map f i is smooth with compact support, we have for any i ∈ N:

f i • (I + θ) -f i L 1 (R n ,R) = R n |f i [x + θ (x)] -f i (x) |dx f i C 0,1 (R n ,R) θ ∞ L n (supp f i ) .
Combining the triangle inequality with these two observations, we deduce that for any i ∈ N:

f • (I + θ) -f L 1 (R n ,R) (1 + 2 n n!) f -f i L 1 (R n ,R) + f i C 0,1 (R n ,R) θ 1,∞ L n (supp f i ) . ( 69
)
Let ε > 0. There exists I ∈ N such that f I -f L 1 (R n ,R) ε 2(1+2 n n!) . We set:

δ := min 1 2 , ε 2 f I C 0,1 (R n ,R) L n (supp f I ) .

For any θ ∈ W 1,∞ such that θ 1,∞ < δ, we get from (69) that f • (I + θ) -f L 1 (R n ,R) ε i.e. the map θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is continuous at the origin. We now assume that f ∈ W 1,1 and we prove that θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is dierentiable at the origin. First, note that the linear map θ ∈ W 1,∞ → ∇f | θ ∈ L 1 (R n , R) is well dened and continuous since we get from the Cauchy-Schwarz inequality:

R n | ∇f (x) | θ (x) |dx R n |∇f (x) | |θ (x) |dx θ 1,∞ ∇f 1 < +∞. (70)
We want to show it is the dierential of θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) at the origin. For this purpose, we introduce the following map:

R f : W 1,∞ -→ L 1 (R n , R) θ -→ R f (θ) := f • (I + θ) -f -∇f | θ .
From the foregoing observations (68) and (70), the map R f is well dened on the open unit ball of W 1,∞ centred at the origin. Therefore, let θ ∈ W 1,∞ be such that θ 1,∞ < 1. Now, let us assume for a moment that f is a smooth map with compact support i.e. f ∈ C ∞ c (R n , R). We consider x ∈ R n and introduce the function ϕ : t ∈ [0, 1] → f [x + tθ(x)]. Since ϕ is the composition of the ane map g x : t ∈ [0, 1] → x + tθ(x) ∈ R n with f , it is dierentiable on [0, 1] and we have:

∀t ∈ [0, 1], ϕ (t) = D t (f • g x ) = D gx(t) f • D t g x = ∇f [g x (t)] | g x (t) = ∇f [x + tθ (x)] | θ (x) .
The Fundamental Theorem of Calculus [33, 7.16] gives 1 0 [ϕ (t) -ϕ (0)]dt = ϕ(1) -ϕ(0) -ϕ (0) thus for any x ∈ R n , we obtain:

1 0 ∇f [x + tθ (x)] -∇f (x) | θ (x) dt = f [x + θ (x)] -f (x) -∇f (x) | θ (x) .

Conclusion

In this article, we have derived various formulas (43) (49) (50) (54) (5) (56) for the shape derivatives of a volume integral, depending on the regularity of the domain. In particular, we have proved that C 1,1 -regularity is enough to dene a notion (6)(7) of partial derivatives with respect to the domain at any order higher than two, while the Lipschitz regularity is enough to dene a shape gradient. Then, we have applied these results in order to obtain the ne shape dierentiability properties associated with (1) and sum up in Table 1. These results have important applications in Quantum Chemistry for the model of Maximal Probability domains (MPDs).

Finally, we conclude by giving some numerical considerations associated with the formula [START_REF] Goudsmit | La découverte du spin de l'électron[END_REF], which is a priori computable by Quantum-Monte-Carlo methods. However, we have seen that it can also involve integrals (2)(3) on the boundary of a domain, which has zero measure from a probabilistic point of view. However, the shape gradient [START_REF] Menéndez | A view of covalent and ionic bonding from maximum probability domains[END_REF], shape Hessian [START_REF] Savin | On the signicance of ELF basins[END_REF], and the kernel (34) of p ν are (n-1)-or (n-2)-dimensional volume integrals. In particular, they can be reasonably computed by Quantum-Monte-Carlo methods. In a future work, we will consider the specic case of wave functions given by a sum of Slater determinants with some numerical applications.

  where (•) ∂Ω := (•) -(•) n n Ω refers to the tangential component of a vector eld, and in particular∇ ∂Ω (•) := ∇(•) -∇(•) | n Ω n Ω is the tangential component of the gradient operator, where II Ω (•, •) := -D ∂Ω n Ω (•) | (•)is the second fundamental form associated to the C 1,1 -surface ∂Ω, which is a symmetric bilinear form on the tangent space, withD ∂Ω (•) := D(•) -D(•)n Ω [n Ω ] Tdenoting the tangential component of the dierential operator on vector elds. In particular, if the perturbations θ and θ are normal to the boundary ∂Ω i.e. if θ ∂Ω = θ∂Ω = 0, then Z[θ, θ] ≡ 0 and the last term in (3) is again equal to zero in this case.

  by standard approximating arguments. Indeed, for any θ ∈ W 1,∞ ∩ C 1 , there exists a sequence (θ k ) k∈N of elements in W 2,∞ such that θ k and [D • θ k ] ij respectively converges to θ and [D • θ] ij uniformly on any compact subset of R n as k → +∞ and for any (i, j) ∈ 1, n 2 (consider again the usual mollier [17, Section 4.2.1 Theorem 1]).

[ 22 ,

 22 Chapter 5] [40, Chapter 2].

  ,∞ -weakly-star, and uniformly on compact sets (consider the usual mollier [17, Section 4.2.1 Theorem 1]). Finally, for anyθ 0 ∈ W 1,∞ ∩ B 0,1 , the domain (I + θ 0 )(Ω) has a Lipschitz boundary ∂[(I + θ 0 )(Ω)] = (I + θ 0 )(∂Ω) and θ • (I + θ 0 ) -1 ∈ W 1,∞ for any θ ∈ W 1,∞ .Hence, (54) can be used to dene (55), concluding the proof of Corollary 3.5.

  we have Def (θ) = Def (I + θ) and I + D x θ = D x (I + θ) for any x ∈ Def (θ). Using again Rademacher's Theorem, the last equality holds true almost everywhere. Finally, let θ ∈ B 0,1 . Proposition 4.1 ensures that (I + θ) has a Lipschitz continuous inverse. Hence, at any point x ∈ A := Def (I + θ) ∩ (I + θ) -1 Def [(I + θ) -1 ] , we can correctly dierentiate the relation (I + θ) -1 • (I + θ) = I and it yields to D (I+θ)(x) [(I + θ) -1 ](I + D x θ) = I. Since we have |||D x θ||| θ 0,1 < 1, the matrix I + D x θ has an inverse [33, 18.3], which is multiplied to the last equality to get D (I+θ)(x) [(I + θ) -1 ] = (I + D x θ) -1 . Combining [17, Section 2.4.1 Theorem 1] and [17, Section 2.2 Theorem 2] with Rademacher's Theorem, we deduce that R n \A has a zero n-dimensional Lebesgue measure i.e. D (I+θ)(x) [(I + θ) -1 ] = (I + D x θ) -1 for almost every x ∈ R n , concluding the proof of Lemma 4.2.

Proposition

  ∀θ ∈ C 0,1 , D 0 (g • f )(θ) = D f (0) g[D 0 f (θ)] = D 0 g[f (θ)] = -f (θ) = -D • θ,concluding the proof of Proposition 4.3.4.3 About the dierentiability related to the Jacobian determinantProposition 4.4. Let θ ∈ B 0,1 . Then, the Jacobian determinant of I + θ i.e. the function x ∈ R n → det[D x (I + θ)] ∈ R is well dened almost everywhere. In addition, it is a measurable map which satises for almost every point x ∈ R n : |det [D x (I + θ)] | n! 1 -θ 0,1 ,

  n 2 is well dened almost everywhere and measurable. Since the determinant is a continuous map and D • (I + θ) = I + D • θ by Lemma 4.2, the Jacobian determinant of I +θ is well dened almost everywhere and measurable. First, we can express the Jacobian determinant of I + θ by using the set S n of permutations of n elements i.e. the set of bijective maps from 1, n into 1, n . Introducing the map s : S n → {-1, 1} dening the signature a permutation, it follows for almost every point x ∈ R n : det [D x (I + θ)] = det (I + D x θ) = p∈Sn s (p) n i=1

  det [D x (I + θ)] = n k=0 I k ⊆ 1,n cardI k =k p∈Sn ∀j / ∈I k ,p(j)=j s (p) i∈I k ∂ i θ p(i) (x) .

2 j

 2 |∂ i θ p(i) (x) | = | [D x θ (e i )] p(i) | n j=1 [D x θ (e i )] = |D x θ (e i ) | |||D x θ||| |e i | θ 0,1 .(66)Combining (65) and (66), we obtain:|det [D x (I + θ)] | n k=0 I k ⊆ 1,n cardI k =k p∈Sn ∀j / ∈I k ,p(j)=j |s (p) | =1 i∈I k |∂ i θ p(i) (x) |

  |det [D x (I + θ)] | n k=0 n(n -1) . . . (n -k + 1) θ k

  ∀θ ∈ C 0,1 , D 0 J(θ) = D 0 [det • (I + f )] (θ) = D I det[D 0 (I + f )(θ)] = trace [f (θ)] div (θ) .

Corollary 4 . 5 . 1 -

 451 Let θ ∈ B 0,1 . Then, the map x ∈ R n → det([D x (I + θ)] -1 ) ∈ R is well dened almost everywhere, measurable, and it satises for almost every point x ∈ R n :|det [D x (I + θ)] -1 | n! (1 -θ 0,1 ) n .(67)In particular, the mapθ ∈ C 0,1 → det([D • (I + θ)] -1 ) ∈ L ∞ (R n , R) is well dened on B 0,1 .Proof. Let θ ∈ B 0,1 . Considering Proposition 4.3 and Lemma 4.2, (I + D • θ) -1 = [D • (I + θ)] -1 is well dened almost everywhere and measurable. Since the determinant is a continuous map, we deduce that x ∈ R n → det([D x (I + θ)] -1 ) ∈ R is well dened almost everywhere and measurable. Applying relation (64) to [D • (I + θ)] -1 , we get for almost every point x ∈ R n : |det [D x (I + θ)] -1 | = |det (I + D x θ) θ 0,1 ) n . Hence, relation (67) holds true and the map θ∈ C 0,1 → det([D • (I + θ)] -1 ) ∈ L ∞ (R n , R)is well dened at any point of B 0,1 , concluding the proof of Corollary 4.5.

  R n |f [x + θ (x)] |dx = R n |f [x + θ (x)] det (I + D x θ) -1 det [D x (I + θ)] |dx n! (1 -θ 0,1 ) n R n |f [x + θ (x)] det (D x (I + θ)) |dx = R n|f (y) |dy < +∞ .

  1,1 . Since the outer normal eld n Ω is now Lipschitz continuous, we deduce from Rademacher's Theorem [17, Section 3.1.2] that it is dierentiable almost everywhere. Hence, we can write:

  1,∞ and (59)(60) hold true. Lemma 4.2. Let θ ∈ C 0,1 . Then, the dierential mapD • θ : x ∈ R n → D x θ ∈ R n 2 is well denedalmost everywhere, measurable, and it is essentially bounded by √ n θ 0,1 in the matrix space R n 2 . In particular, the map θ

= det (I) = det [D x (I + θ)] -1 • D x (I + θ) = det [D x (I + θ)] -1 det [D x (I + θ)] .

Acknowledgements

This work started in January 2015 as a part of project directed by Pascal Frey at the ISCD 1 . The author is very grateful for having given him this opportunity that now sounds as old good times. This work is a part of a larger project nanced by Calsimlab ANR-11-LABX-0037-01 in collaboration with Benoît Braïda and Andreas Savin from the LCT 2 , and also with Eric Cancès from the CERMICS 3 . The nal form of the paper grew while the author was under a postdoctoral position of the CMM 4 .

Theorem 3.6. Let n 2, k 0 2, and f ∈ W k0,1 (R n , R). We consider an open bounded set Ω ⊂ R n with a boundary of class C 1,1 . If we assume that all the vector elds are normal to the boundary i.e. if θ j = (θ j ) n n Ω on ∂Ω for any j ∈ 1, k 0 , then the results of Theorem 3.3 hold true but [START_REF] Brezis | Functional analysis, Sobolev spaces and partial dierential equations[END_REF] [START_REF] Bucur | Anatomy of the shape Hessian via lie brackets[END_REF] can be used instead of (50) to dene D k 0 F (I+θ0)(Ω) in (51). Proof. Let n 2, k 0 2, f ∈ W k0,1 (R n , R), and consider a C 1,1 -domain Ω ⊂ R n . First, assuming that the vector elds are normal to the boundary, we can apply Theorem 3.3: for any k ∈ 1, k 0 , the k-th-order shape derivative of ( 40) is well dened by (50). Since n Ω is a Lipschitz continuous map, it is dierentiable almost everywhere using Rademacher's Theorem [START_REF] Evans | Measure theory and ne properties of functions[END_REF]Section 3.1.2]. Consequently, we can correctly insert the expression (θ j ) n n Ω in the term D ∂Ω θ j of (50). Then, we expand the corresponding product as follows:

Inserting the above expansion in (50), we now distinguish two cases, the last one being itself splitted into two subcases. First, we assume that there exists j ∈ J m such that p(j) ∈ J m . In this case, we can consider the sum involving the indice i p(j) and we get that the term will involve

0 so it is equal to zero. Hence, it only remains terms such that p(J m ) ⊆ I l \J m . Similarly, we can consider two disjoint subcases. If there exists j ∈ I l \J m such that p(j) ∈ J m , then we get that such terms will involve a sum

the last equality coming from the (tangential) dierentiation of the relation |n Ω | 2 = 1 in the local parametrization, and the fact that D ∂Ω n Ω is a self-adjoint endomorphism. Therefore, it only remains the terms for which p(J m ) ⊆ I l \J m and p(I l \J m ) ⊆ I l \J m . We obtain that it remains only the case p(I l ) = I l \J m which is possible only if J m = ∅ i.e. if m = 0. Finally, one can check that this term is precisely the one given in (5) [START_REF] Bucur | Anatomy of the shape Hessian via lie brackets[END_REF], concluding the proof of Theorem 3.6. Corollary 3.7. Consider the assumptions of Theorem 3.6 in the case k 0 = k = 2. Then, the map [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] is twice shape dierentiable at Ω and its second-order shape derivative is given by the following continuous bilinear form:

where ∂F ∂Ω and ∂ 2 F ∂Ω 2 are given in [START_REF] Cancès | How electrons guard the space: shape optimization with probability distribution criteria[END_REF], and where Z[θ, θ] is dened by (4). Moreover, the map 1 and its second-order dierential is given by the following continuous map:

where

is the second-order shape derivative of [START_REF] Sokolowski | Introduction to shape optimization: shape sensitivity analysis[END_REF] at (I + θ 0 )(Ω) given by (56) . Proof. Applying Theorem 3.3, we get that 1 and its second-order dierential is well dened by (57) but the second-order shape derivative of ( 40) is given by:

We can now distinguishing the tangential and normal parts of the operators and of the vector elds, which is allowed because ∂Ω has C 1,1 -regularity. Then, we can apply the Divergence Theorem for surfaces [27, Theorem 6.10], which is valid with C 1,1 -regularity (adapt for example the proofs of [22, Proposition 5.4.9]). We deduce that the second-order shape derivative of (40) takes the form given in (56). Finally, for any

Hence, (56) can be used to dene (57), concluding the proof of Corollary 3.7.

Then, it comes successively:

Hence, using the Fubini-Tonelli Theorem [33, 8.8 Theorem], we have established that:

We now assume that f ∈ W 1,1 and we show that (71) still holds true by a density argument. Indeed, there exists a sequence (f i ) i∈N of smooth maps with compact support converging to f in the W 1,1 -norm [5, Theorem 9.2]. Let i ∈ N. We get from (71) applied to f i :

.

On the one hand, we combine relation (68) applied to the maps f i -f and θ, with observation (70) applied to ∇(f i -f ) in order to obtain:

On the other hand, we combine relation (68) applied to ∇(f i -f ) and tθ in order to get:

Therefore, from these two last inequalities, we deduce that:

By letting i → +∞, we have obtained that relation (71) holds true for any f ∈ W 1,1 . Finally, it only remains to prove that | R f (θ)| → 0 as θ 1,∞ → 0 in order to conclude about the dierentiability of the map θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) at the origin. Again, we are using a density argument. Let f ∈ W 1,1 and θ ∈ W 1,∞ be such that θ 1,∞ 1 2 . There exists a sequence (f i ) i∈N of smooth maps with compact support converging to f in W 1,1 . Let i ∈ N. As before, we get from the triangle inequality and relation (68) applied to the maps ∇(f i -f ) and tθ:

Moreover, since f i is smooth with compact support, we have:

Let ε > 0. There exists I ∈ N such that f I -f 1,1 ε 2(1+2 n n!) . We set:

Consequently, for any θ ∈ W 1,∞ such that θ 1,∞ < δ, we have obtained | R f (θ)| ε as required.

To conclude, | R f (θ)| → 0 as θ 1,∞ → 0 so the map θ ∈ W 1,∞ → f • (I + θ) ∈ L 1 (R n , R) is dierentiable at the origin, and its dierential is given by θ ∈ W 1,∞ → ∇f | θ ∈ L 1 (R n , R).