**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.

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