Calculus of Variations and Geometric Measure Theory

G. De Philippis - B. Velichkov

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

created by velichkov on 05 Mar 2013
modified by dephilipp on 30 Oct 2017


Published Paper

Inserted: 5 mar 2013
Last Updated: 30 oct 2017

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

ArXiv: 1303.0968 PDF


In this paper we prove that the shape optimisation problem'

\[ \min\Big\{\lambda_k(\Omega): \Omega\subset\mathbb R^d \Omega\ \hbox{open},\ P(\Omega)=1,\
<+\infty\Big\}, \] 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.