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

[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.


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