Existence and regularity of minimizers for some spectral functionals with perimeter constraint

created by velichkov on 05 Mar 2013
modified on 23 Apr 2014

[BibTeX]

Published Paper

Inserted: 5 mar 2013
Last Updated: 23 apr 2014

Journal: Appl. Math. Optim.
Volume: 69
Number: 2
Pages: 199--231
Year: 2014

Abstract:

In this paper we prove that the shape optimization problem

$\min\left\{\lambda_k(\Omega):\ \Omega\subset\mathbb{R}^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\ \Omega <+\infty\right \},$

has a solution for any $k\in\mathbb{N}$ and dimension $d$. Moreover, every solution is a bounded connected open set with boundary which is $C^{1,\alpha}$ outside a closed set of Hausdorff dimension $d-8$. Our results are more general and apply to spectral functionals of the form $f(\lambda_{k_1}(\Omega),\dots,\lambda_{k_p}(\Omega))$, for increasing functions $f$ satisfying some suitable bi-Lipschitz type condition.