Abstract. We discuss the stability of "critical" or "equilibrium" shapes of a shape-dependent energy functional. We analyze a problem arising when looking at the positivity of the second derivative in order to prove that a critical shape is an optimal shape. Indeed, often when positivity -or coercivityholds, it does for a weaker norm than the norm for which the functional is twice differentiable and local optimality cannot be a priori deduced. We solve this problem for a particular but significant example. We prove "weak-coercivity" of the second derivative uniformly in a "strong" neighborhood of the equilibrium shape.Résumé. Nous nous intéressonsà la stabilité des formes critiques ou d'"équilibre" d'uneénergie dépendant de la forme. Dans le but de montrer qu'une forme critique est une forme optimale, nouś etudions la positivité de la dérivée seconde. En effet, quand elle a lieu, la coercivité n'est vraie que dans une norme plus faible que celle pour laquelle l'énergie est différentiable: l'optimalité locale ne peut donc pas enêtre déduite a priori. Nous résolvons cette difficulté dans un cas particulier mais néanmoins significatif. Nousétablissons de la "coercivité faible" de la dérivée seconde uniformément dans un voisinage "fort" de la forme d'équilibre.Mathematics Subject Classification. 49Q10, 49K40, 35J25.

We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape function, so that critical stable domains (i.e. such that the first order derivative vanishes and the second order one is positive) are local minima for smooth perturbations; as we are in an infinite dimensional framework, and that in most applications there is a norm-discrepancy phenomenon, this type of result require a lot of work. We show that these hypotheses are satisfied by classical functionals, involving the perimeter, the Dirichlet energy or the first Laplace-Dirichlet eigenvalue. We also explain how we can easily deal with constraints and/or invariance of the functionals. As an application, we retrieve or improve previous results from the existing literature, and provide new local stability results. We finally test the sharpness of our results by showing that the local minimality is in general not valid for non-smooth perturbations.2000MSC : 49K20, 49Q10.Keywords : isoperimetric inequalities, shape optimization, second order sensitivity, stability in shape optimization.One then uses (in the whole paper) the usual notion of Fréchet-differentiability: shape derivatives of J at Ω are the successive derivatives of J Ω at 0, when they exist. In particular, the first shape derivative is J ′ (Ω) := J ′ Ω (0), a continuous linear form on Θ (the shape gradient), and the second order shape derivative is J ′′ (Ω) := J ′′ Ω (0), a continuous symmetric bilinear form on Θ (the shape hessian). • Normal graphs:On the other hand, assuming that Ω is C 1 (and n = n ∂Ω is its outer unit normal vector) we can considerWith the last formula, we easily notice that Q 1 = 0. The sign of Q k can be obtained using [38, section 6.5 page 133] (done when d = 2, but as noticed in [31], valid for any d): indeed, their computations implywhich leads to ∀k ≥ 2, Q k ≥ k − 1. Therefore Theorem 3.2 applies, and we retrieve a Faber-Krahn quantitative inequality in a W 2,p -neighborhood of the ball. Examples with competitionIn this section, B is a ball, X = W 2,p (∂B) for p > d and we denote for η > 0 (see (18) for a definition of d X ):Combining Theorem 3.2 to the computations from Section 2.1, we easily obtain the following result:Proposition 5.1 There exists γ 0 ∈ (0, ∞) such that for every γ ∈ [−γ 0 , ∞), there exists η = η(γ) > 0 and c = c(γ) > 0 such that for every Ω ∈ V η ,Proof of Proposition 5.1: We show that we can apply Theorem 3.2 can be applied to Ω * = B and J ∈ {P + γE, P + γλ 1 , E + γλ 1 , λ 1 + γE)}.It is shown in Sections 3.2 and 4 that (P, E, λ 1 ) satisfy (C H s 2 ) and (IT H s 2 ,X ) for suitable values of s 2 , and with Lemmata 2.9 and 2.8 we easily check that the ball is a critical and strictly stable domain for J under volume constraint and up to translations, either if γ ≥ 0 or if γ < 0 is small enough.Corollary 5.2 With the same notations as in Proposition 5.1, we have, with η 0 = η(γ ...

The paper presents a theoretical study of an identification problem by shape optimization methods. The question is to detect an object immersed in a fluid. Here, the problem is modeled by the Stokes equations and treated as a nonlinear least-squares problem. We consider both the Dirichlet and Neumann boundary conditions. Firstly, we prove an identifiability result. Secondly, we prove the existence of the first-order shape derivatives of the state, we characterize them and deduce the gradient of the least-squares functional. Moreover, we study the stability of this setting. We prove the existence of the second-order shape derivatives and we give the expression of the shape Hessian. Finally, the compactness of the Riesz operator corresponding to this shape Hessian is shown and the ill-posedness of the identification problem follows. This explains the need of regularization to numerically solve this problem.

No abstract

The presence of small inclusions modifies the solution of the Laplace equation posed in a reference domain Ω 0 . This question has been deeply studied for a single inclusion or well separated inclusions. We investigate the case where the distance between the holes tends to zero but remains large with respect to their characteristic size. We first consider two perfectly insulated inclusions. In this configuration we give a complete multiscale asymptotic expansion of the solution to the Laplace equation. We also address the situation of a single inclusion close to a singular perturbation of the boundary ∂Ω 0 . We also present numerical experiments implementing a multiscale superposition method based on our first order expansion.

International audienceThe aim of our work is to reconstruct an inclusion ω immersed in a fluid flowing in a larger bounded domain Ω via a boundary measurement on ∂Ω. Here the fluid motion is assumed to be governed by the Stokes equations. We study the inverse problem of reconstructing ω thanks to the tools of shape optimization by minimizing a Kohn-Vogelius type cost functional. We first characterize the gradient of this cost functional in order to make a numerical resolution. Then, in order to study the stability of this problem, we give the expression of the shape Hessian. We show the compactness of the Riesz operator corresponding to this shape Hessian at a critical point which explains why the inverse problem is ill-posed. Therefore we need some regularization methods to solve numerically this problem. We illustrate those general results by some explicit calculus of the shape Hessian in some particular geometries. In particular, we solve explicitly the Stokes equations in a concentric annulus. Finally, we present some numerical simulations using a parametric method. © 2013 American Institute of Mathematical Sciences

Ventcel boundary conditions are second order differential conditions that appear in asymptotic models. Like Robin boundary conditions, they lead to wellposed variational problems under a sign condition of a coefficient. Nevertheless situations where this condition is violated appeared in several works. The wellposedness of such problems was still open. This manuscript establishes, in the generic case, the existence and uniqueness of the solution for the Ventcel boundary value problem without the sign condition. Then, we consider perforated geometries and give conditions to remove the genericity restriction.

The paper extends investigations of identification problems by shape optimization methods for perfectly conducting inclusions to the case of perfectly insulating material. The Kohn and Vogelius criteria as well as a tracking type objective are considered for a variational formulation. In case of problems in dimension two, the necessary condition implies immediately a perfectly matching situation for both formulations. Similar to the perfectly conducting case, the compactness of the shape Hessian is shown and the illposedness of the identification problem follows. That is, the second order quadratic form is no longer coercive. We illustrate the general results by some explicit examples and we present some numerical results.