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Chapitre D'ouvrage Année : 2017

Regularity of optimal spectral domains

Résumé

In this paper, we review known results and open problems on the question of {\em 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 $\mathbb{R}^d$, $ 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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hal-01322817 , version 1 (27-05-2016)

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  • HAL Id : hal-01322817 , version 1

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Jimmy Lamboley, Michel Pierre. Regularity of optimal spectral domains. Shape optimization and Spectral theory, pp.29-77, 2017, 978-3-11-055088-7. ⟨hal-01322817⟩
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