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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

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.

This paper is devoted to the analysis of a second order method for recovering the a priori unknown shape of an inclusion ω inside a body Ω from boundary measurement. This inverse problem -known as electrical impedance tomography -has many important practical applications and hence has focussed much attention during the last years. However, to our best knowledge, no work has yet considered a second order approach for this problem. This paper aims to fill that void: we investigate the existence of second order derivative of the state u with respect to perturbations of the shape of the interface ∂ω, then we choose a cost function in order to recover the geometry of ∂ω and derive the expression of the derivatives needed to implement the corresponding Newton method. We then investigate the stability of the process and explain why this inverse problem is severely ill-posed by proving the compactness of the Hessian at the global minimizer.

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We consider the question of giving an upper bound for the first nontrivial eigenvalue of the Wentzell-Laplace operator of a domain Ω, involving only geometrical information. We provide such an upper bound, by generalizing Brock's inequality concerning Steklov eigenvalues, and we conjecture that balls maximize the Wentzell eigenvalue, in a suitable class of domains, which would improve our bound. To support this conjecture, we prove that balls are critical domains for the Wentzell eigenvalue, in any dimension, and that they are local maximizers in dimension 2 and 3, using an order two sensitivity analysis. We also provide some numerical evidence.

We aim to reconstruct an inclusion ω immersed in a perfect fluid flowing in a larger bounded domain via boundary measurements on ∂ . The obstacle ω is assumed to have a thin layer and is then modeled using generalized boundary conditions (precisely Ventcel boundary conditions). We first obtain an identifiability result (i.e. the uniqueness of the solution of the inverse problem) for annular configurations through explicit computations. Then, this inverse problem of reconstructing ω is studied, thanks to the tools of shape optimization by minimizing a least-squares-type cost functional. We prove the existence of the shape derivatives with respect to the domain ω and characterize the gradient of this cost functional in order to make a numerical resolution. We also characterize the shape Hessian and prove that this inverse obstacle problem is unstable in the following sense: the functional is degenerate for highly oscillating perturbations. Finally, we present some numerical simulations in order to confirm and extend our theoretical results.

We consider a quantum particle in an infinite square potential well of R n , n = 2, 3, subjected to a control which is a uniform (in space) electric field. Under the dipolar moment approximation, the wave function solves a PDE of Schrödinger type. We study the spectral controllability in finite time of the linearized system around the ground state. We characterize one necessary condition for spectral controllability in finite time: (Kal) if Ω is the bottom of the well, then for every eigenvalue λ of − D Ω , the projections of the dipolar moment onto every (normalized) eigenvector associated to λ are linearly independent in R n . In 3D, our main result states that spectral controllability in finite time never holds for one-directional dipolar moment. The proof uses classical results from trigonometric moment theory and properties about the set of zeros of entire functions. In 2D, we first prove the existence of a minimal time T min (Ω) > 0 for spectral controllability, i.e., if T > T min (Ω), one has spectral controllability in time T if condition (Kal) holds true for (Ω) and, if T < T min (Ω) and the dipolar moment is one-directional, then one does not have spectral controllability in time T . We next characterize a necessary and sufficient condition on the dipolar moment 3917 insuring that spectral controllability in time T > T min (Ω) holds generically with respect to the domain. The proof relies on shape differentiation and a careful study of Dirichlet-to-Neumann operators associated to certain Helmholtz equations. We also show that one can recover exact controllability in abstract spaces from this 2D spectral controllability, by adapting a classical variational argument from control theory.