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.
Domaines
Optimisation et contrôle [math.OC]| Origine | Fichiers produits par l'(les) auteur(s) |
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