# Regularity of the optimal shapes for the first eigenvalue with volume and inclusion constraints

created by lamboley on 21 May 2016

[BibTeX]

Published Paper

Inserted: 21 may 2016
Last Updated: 21 may 2016

Journal: Annales de l'IHP, Analyse non linéaire
Volume: 26
Number: 4
Pages: 1149-1163
Year: 2009

Abstract:

We consider the well-known following shape optimization problem: $\lambda_1(\Omega^*)=\min_{\stackrel{ \Omega =a} {\Omega\subset{ D}}} \lambda_1(\Omega),$ where $\lambda_1$ denotes the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary condition and $D$ is an open bounded set (a box). It is well-known that the solution of this problem is the ball of volume $a$ if such a ball exists in the box $D$ (Faber-Krahn's theorem).\\ In this paper, we prove regularity properties of the boundary of the optimal shapes $\Omega^*$ in any case and in any dimension. Full regularity is obtained in dimension 2.