Regularity of Optimal Spectral domains

created by lamboley on 06 Jun 2016

[BibTeX]

Inserted: 6 jun 2016
Last Updated: 6 jun 2016

Year: 2016
Notes:

Chapter of the book Shape Optimization and Spectral Theory edited by Antoine Henrot and published by De Gruyter

Abstract:

In this paper, we review known results and open problems on the question of regularity of the optimal shapes for minimization problems of the form

$\min\left\{\lambda_k(\Omega), \;\Omega\subset D, \Omega =a\right\}$

where $D$ is an open set in $R^d$, $a\in(0, D )$, $k\in \mathbb{N}^*$ and $\lambda_k(\Omega)$ denotes the $k$-th eigenvalue of the Laplacian with homogeneous Dirichlet boundary conditions. We also discuss some related problems involving $\lambda_k$, but leading to singular optimal shapes. This text is a reproduction of the third chapter of the book “Shape optimization and Spectral theory” (De Gruyter) edited by A. Henrot.