Calculus of Variations and Geometric Measure Theory
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J. Lamboley - M. Pierre

Regularity of Optimal Spectral domains

created by lamboley on 06 Jun 2016


Inserted: 6 jun 2016
Last Updated: 6 jun 2016

Year: 2016

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


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,

where $D$ is an open set in $R^d$, $a\in(0,
)$, $k\in \mathbb{N}^*$ 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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