Antoine Henrot

Minimization problems for eigenvalues of the Laplacian

This paper is a survey on classical results and open questions about minimization problems concerning the lower eigenvalues of the Laplace operator. After recalling classical isoperimetric inequalities for the two first eigenvalues, we present recent advances on this topic. In particular, we study the minimization of the second eigenvalue among plane convex domains. We also discuss the minimization of the third eigenvalue. We prove existence of a minimizer. For others eigenvalues, we just give some conjectures. We also consider the case of Neumann, Robin and Stekloff boundary conditions together with various functions of the eigenvalues.

On some recent advances in shape optimization

Abstract In this Note we give a short review on recent developements in shape optimization. We explain how the generic non-existence of solutions can be circumvent. Either one can impose some geometric restrictions on the class of admissible domains to get existence (we then explain how to write the usual optimality conditions), or generalized designs are allowed which leads to relaxation by homogenization techniques (we thus obtain topology optimization methods).

The one phase free boundary problem for the p-Laplacian with non-constant Bernoulli boundary condition

Our objective, here, is to generalize our earlier results on the existence of classical convex solution to a free boundary problem with a Bernoulli-type boundary gradient condition and with the p-Laplacian as the governing operator. The main theorems of this paper assert that the exterior and the interior free boundary problem with a Bernoulli law, i.e. with a prescribed pressure a(x) on the free streamline of the flow, have convex solutions provided the initial domains are convex. The continuous function a(x) is subject to certain convexity properties. In our earlier results we have considered the case of constant a(x). In the lines of the proof of the main results we also prove the semi-continuity (up to the boundary) of the gradient of the p-capacitary potentials in convex rings, with C-1 boundaries.

Some overdetermined boundary value problems with elliptical free boundaries

In this paper we study three different overdetermined boundary value problems in R2: a problem of torsion, a problem of electrostatic capacity, and a problem of polarization. In each case we prove that a solution exists if and only if the free boundary is an ellipse. The techniques we use rely on classical complex function theory, maximum principle, and some topological argument.

Optimal location of the actuator for the pointwise stabilization of a string

We study the large time behavior of the solutions of a homogenous string equation with a homogenous Dirichlet boundary condition at the left end and a homogenous Neuman boundary condition at the right end. A pointwise interior actuator gives a linear viscous damping term. We give a complete characterization of the positions of the actuator for which the system becomes exponentially stable in the energy space. In the case of nonexponential decay in the energy space we give explicit polynomial decay estimates valid for regular initial data. Moreover we show that the fastest decay rate is obtained if the actuator is located at the middle point of the string. 1. Introduction and main results In this paper we study the asymptotic behaviour of the solution of the equation modelling the vibrations of a string with pointwise damping. More precisely we consider the following initial and boundary value problem: ∂ 2 u ∂t 2 (x, t) − ∂ 2 u ∂x 2 (x, t) + ∂u ∂t (ξ, t)δξ = 0, 0 0,

What is the Optimal Shape of a Pipe

We consider an incompressible fluid in a three-dimensional pipe, following the Navier–Stokes system with classical boundary conditions. We are interested in the following question: is there any optimal shape for the criterion “energy dissipated by the fluid”? Moreover, is the cylinder the optimal shape? We prove that there exists an optimal shape in a reasonable class of admissible domains, but the cylinder is not optimal. For that purpose, we define the first order optimality condition, thanks to the adjoint state and we prove that it is impossible that the adjoint state be a solution of this over-determined system when the domain is the cylinder. At last, we show some numerical simulations for that problem.

Some results about Schiffer's conjectures

We study two overdetermined problems in spectral theory, about the Laplace operator. These problems are known as Schiffers conjectures and are related to the Pompeiu problem. We show the connection between these problems and the critical points of the functional eigenvalue with a volume constraint. We use this fact, together with the continuous Steiner symmetrization, to give another proof of Serrins result for the first Dirichlet eigenvalue. In two dimensions and for a general simple eigenvalue, we obtain different integral identities and a new overdetermined boundary value problem.

On a class of overdetermined eigenvalue problems

In this paper we present some new results of symmetry for inhomogeneous Dirichlet eigenvalue problems overdetermined by a condition involving the gradient of the first eigenfunction on the boundary. One specificity of the problem studied is the dependence of the equation and the boundary condition on the distance to the origin. The method of investigation is based on the use of continuous Steiner symmetrization together with some domain derivative tools. An application is given to the study of an overdetermined eigenvalue problem for a wedge-like membrane.

A new isoperimetric inequality for the elasticae

For a smooth curve

A Shape Optimization Problem for the Heat Equation

\gamma