Existence of surfaces optimizing geometric and PDE shape functionals under reach constraint

This article deals with the existence of hypersurfaces minimizing general shape functionals under certain geometric constraints. We consider as admissible shapes orientable hypersurfaces satisfying a so-called reach condition, also known as the uniform ball property, which ensures C 1,1 regularity of the hypersurface. In this paper, we revisit and generalise the results of [9, 4, 5]. We provide a simpler framework and more concise proofs of some of the results contained in these references and extend them to a new class of problems involving PDEs. Indeed, by using the signed distance introduced by Delfour and Zolesio (see for instance [7]), we avoid the intensive and technical use of local maps, as was the case in the above references. Our approach, originally developed to solve an existence problem in [12], can be easily extended to costs involving different mathematical objects associated with the domain, such as solutions of elliptic equations on the hypersurface.

1 Framework and main results

Introduction
In this paper, we are interested in the question of the existence of optimal sets for shape optimization problems involving surfaces.More precisely, we are interested in shape functionals written as JpΩq " ż BΩ jpx, ν BΩ pxq, B BΩ pxqq dµ BΩ pxq where Ω denotes a smooth subset of R d , the wording 'smooth' being understood at this stage such that all the involved quantities make sense, ν denotes the outward pointing normal vector to BΩ and B BΩ is either a purely geometric quantity such as the mean curvature, or the solution of a PDE on BΩ or on Ω.
We are then interested in the existence of solutions for the optimization problem inf JpΩq .
This kind of problem is very generic.What matters here is that the standard techniques, exposed and developed for example in [7,10], do not apply to d´1 objects and it is necessary to adopt a particular approach.The first question to ask is the choice of the set O ad of all admissible domains.Since the shape functionals we consider involve geometric quantities of the type "outward normal vector to the boundary" or "mean curvature", it is necessary that the manipulated surfaces are not too irregular.For this reason, we choose to impose a constraint that guarantees a uniform regularity, say C 1,1 , of the manipulated sets.This uniform regularity constraint is imposed by using the notion of "reach".Thus, the set O ad represents the set of surfaces having a reach uniformly bounded by below.The precise definition of this notion will be given in Section 1.3.This kind of problem has been the subject of recent studies and results [9,4,5], which have provided positive answers to the existence issues.In their approach, the authors used an efficient, but nevertheless laborious, approach based on the parametrization of the manipulated surfaces, seen as regular manifolds, using local charts.
The objective of this paper is to promote a different approach, based on the extension of the functions defined on the manipulated surfaces to volume neighborhoods, the introduction of an extruded surface and the rewriting of the surface integrals as volume integrals using ad-hoc variable changes.This is a methodological paper, in which a proof method is presented that may work in many cases.The results contained in the article illustrate this point.We discuss possible generalizations of these results in a concluding section.
This method allows to gain conciseness and provides much shorter and direct existence proofs than in the above references.The method also allows to extend the field of investigation to new families of problems, involving the solution of a PDE defined on a hypersurface.Nevertheless, some arguments used by the authors of [9,4,5] cannot be shortened by using our approach.We have therefore chosen to expose our method in a short article, in which we detail all the parts of the proof that can be condensed and we make the necessary reminders concerning the results that cannot be condensed.
The article is organized as follows: we introduce the definition of the reach of a surface as well as the class of admissible sets we will deal with in Section 1.3.The main results of this article, regarding several existence results for shape optimization problems involving surfaces, are provided in Section 1.4.The whole section 2 is devoted to the proofs of the main results.In these proofs, we detail the arguments based on our approach and leading to simplified proofs of the results in [9,4,5].In order to illustrate the potential of our approach, we also provide an existence result involving a general functional depending on the solution of a PDE on the sought manifold.

Notations
Let us recall some classical notations used throughout this paper: • For the sake of notational simplicity, we will sometime use the notation Γ (resp.Γ n ) to denote the hypersurfaces BΩ (resp.BΩ n ).
• The Euclidean inner product (resp.norm) will be denoted x¨, ¨y (resp.} ¨} or sometimes | ¨| when no confusion with other notations is possible).
• Given two positive integers k ď d and Ω Ă R d , H k pΩq denotes the k-dimensional Hausdorff measure of Ω.
• Given Ω Ă R d , the distance (resp., signed distance) to Ω is defined for all x P R d by • Given Ω Ă R d and h ą 0, the tubular neighborhood U h pΩq is defined as Recall that, if BΩ is a nonempty compact C 1,1 -hypersurface of R d , then there exists h ą 0 such that Ω satisfies a uniform ball condition, namely where B h pxq stands for the open ball of radius h centered in x.Furthermore, assuming H d pBΩq " 0, we have the simpler characterization Conversely, if BΩ is nonempty and satisfies Condition (B h ), then its reach is larger than h and the Lebesgue measure of BΩ in R n is equal to 0. Furthermore, BΩ is a C 1,1 hypersurface of R n .We refer for instance to Theorems 2.6 and 2.7 in [4].
• For a given oriented C 1,1 -hypersurface BΩ, we denote by ∇ BΩ or ∇ Γ the tangential gradient and by ∇ the full gradient in R d .When needed, each gradient will be assimilated to a line vector in R d .
• S d´1 denotes the unit sphere of R d .
• M d pRq denotes the linear space of d ˆd matrices with real entries, endowed with the Euclidean operator norm } ¨}.Id denotes the identity matrix in R d .
• For a given C 1,1 hypersurface BΩ, we denote by H BΩ : BΩ Ñ R, its mean curvature.We refer to Appendix A for proper definitions.

Preliminaries on sets of uniformly positive reach
Given r 0 ą 0 and a nonempty compact set D Ă R d , let us introduce the set O r0 of admissible shapes whose reach is bounded by r 0 , namely The elements of O r0 are known to satisfy the following properties.
2. For x P BΩ, ∇b Ω pxq is the unit outward normal vector.
3. For h ă r 0 , ∇b Ω is 2 r0´h -Lipschitz continuous on the tubular neighborhood U h pBΩq.We will endow the set O r0 with a 'sequential' topology, by introducing a notion of convergence in this set.

Definition 1 (R-convergence in O r0
).Given pΩ n q nPN P O N r0 , we say that pΩ n q nPN R-converges to Ω 8 P O r0 and we write α pU r pBΩ 8 qq, @r ă r 0 , @α P r0, 1q, weakly-star in W 2,8 pU r pBΩ 8 qq, @r ă r 0 . (2) The next result justifies the interest of the class O r0 endowed with the R-convergence for existence issues.

Proposition 1. O r0 is sequentially compact for the R-convergence.
The proof of this proposition can be found in Section B. Let us end this section by providing several additional properties of the R-convergence.
2. H d pΩ n q converges toward H d pΩ 8 q as n Ñ `8.
3. If all the BΩ n belong to the same isotopic class, then BΩ 8 also belongs such a class.
The proof of this lemma is given in Section 2.2.

Remark 1.
According to Lemma 2, we obtain for example that for a given Ω 0 P O r0 , and a ď b,

Main results
Let us introduce the general shape functional where j 1 is continuous from R d ˆSd´1 ˆR to R and convex with respect to its last variable.We recall that ν and H BΩ denote respectively the outward pointing normal vector and the mean curvature.
According to Proposition 1, the set O r0 is sequentially compact for the R-convergence.Therefore, in order to infer the existence of an optimal surface minimizing F 1 over O r0 it is enough to prove the lower semicontinuity of functional F 1 (under suitable assumptions on the function j 1 ).This is the main purpose of the following result.
Theorem 1 ([4], Theorem 1.3).Let us assume that j 1 is continuous with respect to all variables and convex with respect to its last one.Then, F 1 is a lower semi-continuous shape functional for the R-convergence, i.e., for every sequence pΩ n q nPN P O N r0 that R-converges toward Ω 8 , one has As a consequence, the shape optimization problem It is notable that, by applying Theorem 1 both to j 1 and ´j1 , we get the following corollary.
Corollary 1.If j 1 is continuous and linear in the last variable, then F 1 is a continuous shape functionals for the R-convergence.
Remark 2. In the case where d " 3, it is proved in [4,Theorem 1.3] that Theorem 1 holds if we replace the mean curvature by the Gaussian one in the definition of F 2 .We do not provide a proof here since most of the difficulties are related to the convergence of a product of weak-star converging sequences and our approach does not change the proof in a significant way.
Let us now consider two classes of shape optimization problems involving either an elliptic PDE inside Ω or an elliptic PDE on the C 1,1 hypersurface BΩ.
Problems involving an elliptic PDE on a C 1,1 -hypersurface of R d .Given f P C 0 pDq, we consider the problem of minimizing a shape functional depending on the solution v BΩ of the equation where ∆ BΩ denotes the Laplace-Beltrami operator on BΩ.Since we are not considering C 8 manifolds but rather C 1,1 ones, we need to explain how the PDE must be understood.We use here an energy formulation, defining, for a closed and nonempty hypersurface BΩ, the functional where H 1 ˚pBΩq denotes the Sobolev space of functions in H 1 pBΩq with zero mean on BΩ.We hence define v BΩ as the unique solution of the minimization problem min The proof of this result is postponed to Section C. Let us introduce the shape functional where j 2 : R d ˆSd´1 ˆR ˆRd Ñ R is assumed to be continuous.
Theorem 2. The shape functional F 2 is lower semi-continuous for the R-convergence, i.e., for every sequence As a consequence, the shape optimization problem Problems involving an elliptic PDE in a domain of R d .Finally, let us investigate the case of a shape criterion involving the solution of a PDE on a domain of R d .We consider hereafter a Poisson equation with non-homogeneous boundary condition, but we claim that all conclusions can be easily extended to a larger class of elliptic PDEs.Let h P L 2 pDq, g P H 2 pDq, and define u Ω as the solution of Let us introduce the shape functional F 3 given by where j 3 : R d ˆSd´1 ˆR ˆRd Ñ R is continuous.
We mention this theorem demonstrated in [5].Nevertheless, it is notable that by adapting the proof of Theorem 2, it is possible to obtain a much shorter proof of this theorem.In order not to make this article unnecessarily heavy, we only give the main steps of the proof in Section 2.5.This example is mentioned both for the sake of completeness, in order to review the existing literature, and also to underline the potential of the approach introduced here, which allows to find more direct proofs of all the known results and to extend them.
In addition, it is interesting to notice that our approach allows to deal with problems involving PDEs both using weak formulations as in (9) and also whose solutions are obtained using a minimization principle, as is the case in (5).The approach thus seems robust and we believe that it can be easily adapted to general families of problems (for example to a general non-degenerate elliptic PDE).

The extruded surface approach
One of the key ideas to prove sequential continuity of functionals involving an integral on the boundary is to approximate such an integral by an integral on a small tubular neighborhood (as done e.g. in [6]).
Let us first illustrate the method by proving Point 5 of Lemma 1.
Remark 3. Note that as ∇b Ω px 0 q is a normal unit vector to BΩ at x 0 , we can identify the tangent hyperplane T x0 BΩ with R d´1 endowed with an Euclidean structure inherited from that of R d .We will use this identification several times in this paper.
As a result, we can identify R ˆTx0 BΩ Q ps, yq Þ Ñ y `s∇b Ω px 0 q with an orthogonal matrix.Moreover, up to the choice of a different orientation on T x0 BΩ, such a matrix belongs to the special orthogonal group SOpnq.We use the same coordinate representation to identify R ˆTx0 BΩ Q ps, yq Þ Ñ d x0 ∇b Ω pyq with a n ˆn matrix.By uniform continuity of the determinant around SOpdq, there exists C 0 ą 0 such that, for every M P SOpdq and every l P M d pRq such that }l} ď C 0 , As ∇b Ω is 2 r0 -Lipschitz continuous on BΩ, we have that for almost every x 0 P BΩ and every Let us fix h ă minpr 0 , r 0 C 0 {2q (independent of Ω), so that }t 0 d x0 ∇b Ω } ď C 0 for almost every x 0 P BΩ and every t 0 P p´h, hq.By the change of variable formula we then have whence the conclusion.

Extruded surface and R-convergence
Let us now illustrate the power of this approach in the case of a R-converging sequence.
Let Ω n R Ý Ñ Ω 8 .From now on, we will use the notation Γ n :" BΩ n for the hypersurfaces.For h ă r 0 and n P N, let us define a parametrization of a neighborhood of Γ n by Lemma 4. For every ε ą 0, there exists h ą 0 such that for all n P N, 1 ´ε ď detpd pt0,x0q T n q ď 1 `ε, for a.e.pt 0 , x 0 q P p´h, hq ˆΓn .
Proof.We follow the same argument as in Section 2.1.1.Namely, for a given ε ą 0, there exists C 0 ą 0 such that for every M P SOpdq and every l P M d pRq such that }l} ď C 0 , Let us fix h ă minpr 0 , r 0 C 0 {2q (independent of n).As ∇b Ωn is 2 r0 -Lipschitz continuous on Γ n , we get }t 0 d x0 ∇b Ωn } ď C 0 for almost every x 0 P Γ and every t 0 P p´h, hq.Whence, using Equation (11) with the previous estimate, we conclude the proof.

Remark 4.
In what follows, we will use the Bachmann-Landau notation o hÑ0 p1qfor a function converging to 0 in L 8 as h goes to 0 and for a given n, large enough.For example, Lemma 4 implies that detpdT n q " 1 `ohÑ0 p1q, on p´h, hq ˆΓn , which means @ε ą 0, DN 0 P N, Dh ą 0, @n P N, n ě N 0 implies ˇˇdetpd pt0,x0q T n q ´1ˇˇď ε, for a.e.pt, xq P p´h, hq ˆΓn .
Let us now introduce the orthogonal projection p n onto Γ n , defined on U h pΓ n q for every h P p0, r 0 q.
Lemma 5.The following properties hold: 1. p n coincides with the second component of T ´1 n : U h pΓ n q Ñ p´h, hq ˆΓn .2. For all x P U h pΓ n q, p n pxq " x ´bΩn pxq∇b Ωn pxq.We can now state the key equality to relate surface and volume integrals.Apply Lemma 4 with ε P p0, 1q to select h ą 0 such that T n : p´h, hq ˆΓn Ñ U h pΓ n q is invertible for every n P N1 .Lemma 6.For all n P N, f P L 1 pΓ n q, and t P p0, hq we have Proof.Using the change of variable formula (also known as area formula for Lipschitz continuous functions), one gets From now on, we will the T ´1 n pyq inside the determinant to improve the readability.

Lemma 7.
For every h ď r 0 {2 and 0 ă t ă h, there exists N 0 such that @n ě N 0 , U h´t pΓ 8 q Ă U h pΓ n q Ă U h`t pΓ 8 q.
Proof.By uniform convergence of b Ωn toward b Ω8 , we have that for n large enough b ´1 Ω8 ppt ´h, h ´tqq Ă b ´1 Ωn pp´h, hqq Ă b ´1 Ω8 pp´h ´t, h `tqq.
In order to perform changes of variable in surface integrals, it is convenient to use directly p n as a way to map Γ 8 onto Γ n .To this aim, we define Note that for n large enough, Lemma 7 ensure that τ n is well-defined.We also introduce Jacpτ n q to denote the Jacobian of τ n .Then we have the following lemma.This is a contradiction, hence τ n is injective which implies that it is a diffeomorphism from Γ 8 to Γ n .

Proof of Lemma 2
Suppose that pΩ n q nPN P O N r0 R-converges toward Ω 8 P O r0 .

Proof of Point 1
For h ă r 0 and using Lemma 6, we have detpdT n q dy.By Lemma 7, moreover, detpdT n q dy for t P p0, hq and n large enough.Let us compare the first term in the right-hand side with Using Lemma 4, detpdT 8 q " detpdT n q `ohÑ0 p1q on p´h, hq ˆΓ8 .Besides, detpdT n q dy " H d´1 pΓ 8 q `ohÑ0 p1q `O ˆt h ˙.

Proof of Point 2
Using the uniform convergence of b Ωn to b Ω8 , we deduce that for every ε ą 0 there exists Hence, we get By inner regularity of Similarly, by outer regularity , where we used that Ω 8 belongs to O r0 .

Proof of Point 3
We want to prove that Γ n is isotopic to Γ 8 for n large enough.We consider According to Lemma 8, ϕ n p1, ¨q " τ n is a diffeomorphism from Γ 8 onto Γ n .Besides, following the proof of Lemma 8, we easily get that for t P p0, 1q, ϕ n pt, ¨q is a diffeomorphism onto its image.

Proof of Theorem 1
Suppose that pΩ n q nPN P O N r0 R-converges toward Ω 8 P O r0 .Let 0 ă t ă h small enough (to be fixed later) and n large enough.
We recall that the unit normal vector to Γ n is given by ∇b Ωn (see Lemma 5).Then, according to Lemma 6, The key idea is to prove that all arguments of j 1 in the first term convergence toward their analogues for n " 8 and to ensure that the second term is small for t small.Let us start with comparing the first term in the right-hand side with F 1 pΩ 8 q.Notice that detpdT n q detpdT 8 q j 1 pp n pyq, ∇b Ωn pp n pyqq, H Γn pp n pyqqq detpdT 8 q dy.
By Lemma 4, we have Let us now investigate the mean curvature term.Note that this term is slightly technical to handle for two reasons: • the mean curvature H Γn is defined as the trace of the shape operator, which is itself defined as the differential of the restriction to the hypersurface of ∇b Ωn (see A); • the Hessian of b Ωn converges only in a weak sense.
We will use the following lemma, which is obtained thanks to the chain rule.Besides, one has that ∇ 2 b Ωn pτ ´1 n pp n pxqqq exists as well.Notice that the last part of the statement is not explicitly contained in [6] but can be obtained by straightforwardly adapting the proof of its Theorem 4.4.
As ∇ 2 b Ωn is uniformly bounded on a neighboorhood of Γ 8 and that b Ωn pxq ď h for x P U h pΓ n q, there exists C ą 0 such that ess sup xPU h pΓnq }rId ´bΩn pxq∇ 2 b Ωn pxqs ´1 ´Id } ď Ch, for h small.As a consequence, using Lemma A.1 (given in the appendix), one has Note also that H Γn ď 1 r0 on Γ n .We can use the uniform continuity of j 1 on a compact set to ensure that for n large enough and n " 8, weak star in L 8 pU r 0 2 pΓ 8 qq.Thus, using for example [1], we have lim inf In order to conclude, let us check that the term in line (17) is small.Since j 1 is continuous on a compact set, it admits a minimum m 0 P R. Let m 1 " minp0, m 0 q ď 0.Then, " o hÑ0 p1q and ż U h˘t pΓ8q m 1 detpdT 8 q dy " 2ph ˘tqm 1 H d´1 pΓ 8 q, we get Finally, combining Equations ( 19)-( 21), we obtain lim inf nÑ`8 Hence, taking h Ñ 0 while ensuring t " ophq gives lim inf nÑ`8 and finishes the proof.

Proof of Theorem 2
Let pΩ n q nPN denote a sequence that R-converges to Ω 8 , and let v n denote the unique solution v Γn to Problem (7) for Ω " Ω n .The difficult part here is that v Γn is not defined on Γ 8 .Our main tool will be τ n , the restriction to Γ 8 of the orthogonal projection p n on Γ n .Those objects were introduced in Section 2.1.2and we proved that τ n is a diffeomorphism between Γ 8 and Γ n in Lemma 8.
We also have to be careful when we transport the tangential gradient of a function.In order to relate the tangential gradient and the ambient gradient, we establish the following pointwise estimate.Lemma 10.Let n P N and f n P H 1 pΓ n q.Then f n ˝τn P H 1 pΓ 8 q and, for almost every x P Γ 8 , where tangential gradients are understood as d-dimensional line vectors and C n pxq " p∇b Ωn pxq J ∇b Ωn pxq ´Idq∇b Ω8 pxq J ∇b Ω8 pxq `bΩn pxq∇ 2 b Ωn pxqp∇b Ω8 pxq J ∇b Ω8 pxq ´Idq.
Besides, C n converges toward zero in the L 8 norm: Proof.First notice that ∇ Γ8 pf n ˝pn qpxq " ∇pf n ˝pn ˝p8 qpxq for almost every x P BΩ 8 , since the directional derivative of f n ˝pn ˝p8 at the point x in the direction ∇b Ω8 pxq is zero.By Lemma 9, ∇ 2 b Ωn pxq is well-defined for almost every x in Γ 8 .By Lemma 5 and the chain rule we obtain, almost everywhere on Γ 8 , ∇pf n ˝pn ˝p8 qpxq " pp∇f n q ˝pn qpId ´∇b J Ωn ∇b Ωn ´bΩn ∇ 2 b Ωn qpId ´∇b J Ω8 ∇b Ω8 ´bΩ8 ∇ 2 b Ω8 q " pp∇ Γn f n q ˝τn qpId ´pId ´∇b J Ωn ∇b Ωn q∇b J Ω8 ∇b Ω8 ´bΩn ∇ 2 b Ωn pId ´∇b J Ω8 ∇b Ω8 qq, where we used that ∇f n " ∇ Γn f n , p n " τ n , and b Ω8 " 0 on Γ 8 .This shows Eq. ( 22).
Let us now bound the L 8 norm of C n .There exists C ą 0 such that, for every n satisfying Γ 8 Ă U r 0 2 pΓ n q, ess sup xPΓ8 }∇ 2 b Ωn pxqp∇b J Ω8 pxq∇b Ω8 pxq ´Idq} ď C. Besides, }b Ωn } L 8 pΓ8q converges to zero.Finally, using the uniform convergence of ∇b Ωn toward ∇b Ω8 , we get This concludes the proof of Eq. ( 23).
From the solution v n in H 1 ˚pΓ n q, we introduce the function w n defined on Γ 8 by Note that, defined as such, w n belongs to H 1 ˚pΓ 8 q.
Step 1: convergence of pw n q nPN .Let us start by considering the sequence of energies pE Γn pv n qq nPN .This sequence is upper bounded by 0, since E Γn pv n q ď E Γn p0q " 0 for every n.By using the uniform Poincaré inequality stated in Proposition C.1 combined with the Cauchy-Schwarz inequality, we get that p ş Γn |v n | 2 dµ Γn pxqq nPN is bounded.We now compute 1 where we used Lemma 8, the Cauchy-Schwarz inequality, and the fact that v n has zero average on Γ n .Hence, we infer that w n " v n ˝τn `onÑ8 p1q.Besides, by performing a change of variable and by using Lemmas 8 and 10, we get where we used that ∇ Γ8 w n " ∇ Γ8 pv n ˝τn q by definition of w n .By using Proposition C.1 and again the Cauchy-Schwarz inequality, we successively infer that the sequences p ş Γ8 |w n | 2 dµ Γ8 pxqq nPN and p ş Γ8 |∇ Γ8 w n | 2 dµ Γ8 pxqq nPN are bounded.By using Theorem C.1, the sequence pw n q nPN converges up to a subsequence toward w 8 P H 1 ˚pΓ 8 q, weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q.Up to extracting a subsequence, we get As a consequence, E Γ8 pw 8 q ď lim inf nÑ`8 Step 2: Minimality of w 8 .Let u P H 1 ˚pΓ 8 q be given and define z n in H 1 ˚pΓ n q by Let n P N. By minimality, one has E Γn pv n q ď E Γn pz n q.
By mimicking the arguments and computations of the first step, we easily get that yielding at the end E Γ8 pw 8 q ď E Γ8 puq.We infer that w 8 is the unique solution to the variational problem (7).Since the reasoning above holds for any closure point of pw n q nPN , it follows that the whole sequence pw n q nPN converges toward w 8 , weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q.Finally, using u " w 8 in (26), we obtain that E Γ8 pw 8 q " lim inf nÑ`8 E Γn pv n q.
In particular p}w n } 2 H 1 pΓ8q q nPN converges toward }w 8 } 2 H 1 pΓ8q which implies the strong convergence of w n in H 1 pΓ 8 q.
Besides, according to the results above and Lemma 5, the following convergences hold where w 8 is the unique solution to the variational problem (7).
By applying [1, Theorem 1], one has lim inf nÑ`8 This is the desired conclusion.

Main steps in the proof of Theorem 3
First note that u Ω ´g solves Eq. ( 9) with source term h ´∆g and Dirichlet boundary condition.As a consequence, we can reduce our study to the case of homonegeous Dirichlet condition (i.e., u Ω " 0 on Γ).
The method relies on a uniform extension property proved by Chenais in [3] for surfaces satisfying an ε-cone condition, which is weaker than the uniform ball condition.

Lemma 11 ([3, Theorem II.1]
).There exists a positive constant C (depending only on r 0 and D) such that for every Ω P O r0 there exists an extension operator E Ω P LpH 2 pΩq, H 2 pDqq satisfying We will use this lemma to extend the solution of the PDEs to the whole box D. The next step is to find a uniform H 2 estimate of the solutions.In our case such an estimate was proved by Dalphin who extended a result for domains with C 2 boundary obtained by Grisvard in [8].

Lemma 12 ([5, Proposition 3.1]
).There exists C ą 0 (depending only on r 0 and D) such that for every Ω P O r0 and f P H 2 pΩq X H The next step is to prove that the restriction to Ω 8 of u ˚is u Ω8 .
To this aim, let us consider an arbitrary compact set K contained in the interior of Ω 8 and a C 8 function ϕ with compact support included in K.For n large enough, K is contained in the interior of Ω n (see Lemma 7), and, therefore, one has ϕ P H 1 0 pΩ n q for such integers n.Using the variational formulation of the PDE (9), we get ż D x∇E Ωn pu Ωn q, ∇ϕy ´f ϕ " 0. (31) Using the density of C 8 functions with compact support in H 1 0 pΩ 8 q and passing to the limit yields that u ˚|Ω8 " u Ω8 .The last step is to relate F 3 pΩ n q and F 3 pΩ 8 q.Since the involved functions belong to Sobolev spaces and since one aims at comparing surface integrals with tubular ones, we need a suitable uniform trace result.
Lemma 13.There exists C such that for every h ă r0 2 , every n P N and every f P H where f denotes the trace of f on Γ n .
Proof.Let f be a smooth function.According to Lemma 5, every point y P U h pΓ n q can be written in a unique way as y " x `t∇b We conclude thanks to the density of the smooth functions in H 1 .
Using that u Ωn is uniformly bounded in H 2 pDq, let us apply Lemma 13 to u Ωn and ∇u Ωn .We obtain }u Ωn ´uΩn ˝pn } 2 L 2 pU h pΓnqq `}∇u Ωn ´p∇u Ωn q ˝pn } 2 L 2 pU h pΓnqq " o hÑ0 phq.The end of the proof is similar to the one of Theorem 1 and consists in using the extruded surface approach to prove lim inf nÑ`8 F 3 pΩ n q ě p1 `ohÑ0 p1qq lim inf nÑ`8

Conclusion
In this paper, we have introduced a new method to tackle the existence issue for shape optimization problems under uniform reach constraints on the considered shapes, of the type inf ΩPOr 0 ż BΩ jpx, ν BΩ pxq, B BΩ pxqq dµ BΩ pxq.

B R-convergence: proof of Proposition 1
The compactness property follows from two facts.First, the Arzelà-Ascoli theorem, combined with the fact that every function b Ω , for Ω P O r0 , is 1-Lipschitz continuous.Second, the reach constraint which imposes a uniform bound on the second derivative of b Ω .These two facts are used in [7] and [4] to get the sequential compactness results used below.Let pΩ n q nPN denote a sequence in O r0 .By the compactness property of sets of uniformly positive reach proved in [7,Chapter 6], it follows that, up to a subsequence, b Ωn converges to b Ω8 for the C 0 topology on D. In [4] the convergence is shown to hold also for the strong C 1,α topology (for α ă 1) and for the weak W 2,8 topology in a r-tubular neighborhood of BΩ 8 , with r ă r 0 .
C The Laplace-Beltrami equation on a manifold: proof of Lemma 3 Let pBΩ, gq denote a closed compact manifold.We explain hereafter how to understand the equation ∆ BΩ v " h in BΩ in a weak sense, whenever Ω P O r0 .Indeed, under this assumption, BΩ is a C w n pxq dµ Γ8 " 0.
By using the first equality, we get that w ˚is constant on Γ and we obtain a contradiction with the two last equalities above.
Let us now prove Lemma 3. Let pu n q nPN denote a minimizing sequence for Problem (7).Since pE Γ pu n qq nPN is bounded, and since E Γ pu n q ě Cpd, r 0 q}u n } 2 L 2 pΓq ´}h} L 2 pΓq }u n } L 2 pΓq according to Proposition C.1, we infer that p}u n } L 2 pΓq q nPN is bounded.Since ż Γ |∇ Γ u n pxq| 2 dµ Γ " E Γ pu n q `żΓ u n pxqhpxq dµ Γ ď }h} L 2 pΓq }u n } L 2 pΓq , we infer the existence of u ˚P H 1 ˚pΓq such that, up to a subsequence, pu n q nPN converges weakly in H 1 ˚pΓq and strongly in L 2 pΓq.Up to extracting a subsequence, we get inf Ω to BΩ is 1 r0 -Lipschitz continuous. 5.There exists a constant C depending only on d, r 0 , and D such that H d´1 pBΩq ď C. Points 1 and 2 are proved in [7, Theorem 8.2, Chapter 7].Points 3 and 4 are proved in [4, Theorems 2.7 and 2.8].The proof of Point 5 is given in Section 2.1.1.

3 .
p n converges toward p 8 in L 8 pU h pΓ 8 qq.Proof.Items 1 and 2 are obviously equivalent and are proved in [7, Theorem 7.2, Chapter 7].Item 3 follows from the C 1 convergence of b Ωn toward b Ω8 .

Remark 5 .
In order to replace Dirichlet boundary conditions by Neumann's ones, one can follow similar steps as those leading to Equation (30).Then, by considering the variational formulation with ϕ P C 8 pDq and passing to the limit in ż Γn gB ν ϕ Ñ ż Γ8 gB ν ϕ, (consequence of Corollary 1 if g P C 0 pDq) one gets that u ˚|Ω8 " u Ω8 .
j j nÑ8 Tr ∇ 2 b Ω8 1 0 pΩq, we have }f } H 2 pΩq ď C}∆f } L 2 pΩq .(28) As a consequence, we have a uniform H 2 pDq estimate on the extension of the solution u Ω , namely, }E Ω pu Ω q} H 2 pDq ď C}h} L 2 pDq , @Ω P O r0 .Ω 8 .Using Eq. (29), we get that pE Ωn pu Ωn qq nPN is uniformly bounded in H 2 pDq.Up to extracting a subsequence, we can assume that E Ωn pu Ωn q ÝÝÝÑ Ωn pxq with x " p n pyq P Γ n and t P p´h, hq.Moreover, one has |f px `t∇b Ωn pxqq ´f pxq| 2 ď C 2 }B ∇b Ωn pxq f px `y∇b Ωn pxqq} 2 Ωn stands for the derivative in the direction ∇b Ωn pxq and C is the norm of the continuous embedding of H 1 pr´r 0 2 , r0 2 sq into the space C Γn|f px `t∇b Ωn pxqq ´f pxq| 2 detpdT n 3 px, ∇b Ωn pp n pxqq, E Ωn pu Ωn qpxq, ∇E Ωn pu Ωn qpxqq dx 3 px, ∇b Ω8 pp 8 pxqq, u ˚pxq, ∇u ˚pxqq dx j j 1,1 submanifold according to Lemma 1, not necessarily C 2 , which justifies why such an equation cannot be understood in a strong sense.The key ingredient in what follows is the Rellich-Kondrachov lemma, stating the compactness of the embedding H 1 ˚pBΩq ãÑ L 2 pBΩq.BΩ |∇u n pxq| 2 dµ BΩ q nPN is bounded.There exists u ˚P H 1 ˚pBΩq such that, up to a subsequence, pu n q nPN converges to u ˚weakly in H 1 ˚pBΩq and strongly in L 2 pBΩq.Proof.According to [6, Th 4.5.ii],sinceBΩ is C 1,1 , the L 2 norm } ¨}L 2 pBΩq on the surface BΩ and the L 2 norm L 2 pBΩq Q u Þ Ñ }u ˝pΩ } L 2 pU h pBΩqq on the thickened surface U h pBΩq are equivalent whenever h ą 0 is small enough, where p Ω pxq denotes the orthogonal projection of x onto BΩ, that is, p Ω pxq " x ´bΩ pxq∇b Ω pxq, and U h pBΩq " tx P R d | |b Ω pxq| ă h and p Ω pxq P BΩu.Similarly, according to [6, Th 4.7.v],sinceBΩ is C 1,1 , the norm } ¨}H 1 ˚pBΩq defined as|∇ Γ u ˝pΩ | 2 dµ BΩare equivalent whenever h ą 0 is small enough.We conclude by using the standard Rellich-Kondrakov theorem (see e.g.[2, Section 9.3]) on the thickened surface U h pBΩq.The following result is a Poincaré type lemma, uniform with respect to the chosen surface in the set O r0 .Proposition C.1 (Poincaré lemma on a surface).Let r 0 ą 0 and Ω P O r0 .There exists Cpr 0 , Dq ą 0 such that @u P H 1Proof.Let pΩ n , v n q nPN , with v n P H 1 ˚pΓ n q, be a minimizing sequence for the problem inf Γn |v n pxq| 2 dµ Γn " 1 by homogeneity of the Rayleigh quotient.According to Proposition 1, we can assume without loss of generality that pΩ n q nPN R-converges toward Ω 8 P O r0 .Let p n denote the orthogonal projection on Γ n and let us introduce the function w n defined in U h pΓ n q for h as in Lemma 4 and n large enough by w n " v n ˝pn .We follow exactly the same lines as in the first step of the proof of Theorem 2. A direct adaptation of the first step of the proof of Theorem 2 yieldsż Γ8 |∇ Γ8 w n pyq| 2 dµ Γ8 pyq " Γn |∇ Γn v n | 2 dµ Γn pxq `op1q.By using Theorem C.1, we get that the sequence pw n q nPN converges up to a subsequence toward w ˚P H 1 ˚pΓ 8 q weakly in H 1 pΓ 8 q and strongly in L 2 pΓ 8 q.Up to extracting a subsequence, we get ż Γ8 |∇ Γ8 w ˚pxq| 2 dµ Γ8 ď lim inf Theorem C.1 (Rellich-Kondrachov theorem on surfaces).Let Ω P O r0 .Let pu n q nPN denote a sequence in H 1 ˚pBΩq such that p ş Γ |∇ Γ upxq| 2 dµ Γ ě Cpr 0 , Dq ż Γ |upxq| 2 dµ Γ .Γn |∇ Γn v n pxq| 2 dµ Γn ď 1 n and ż