Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

Details Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints Regularity of the optimal shape for the first eigenvalue of the Laplacian with volume and inclusion constraints

2008-07-14

arxiv.org/abs/0807.2196v1

Mathematics

Optimization and Control

Scientific paper

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 Laplace operator 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.

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