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

TL;DR

This paper proves the existence and regularity of minimizers for spectral shape optimization problems with perimeter constraints, extending results to more general spectral functionals and establishing smoothness properties of solutions.

Contribution

It establishes the existence and regularity of minimizers for spectral functionals with perimeter constraints, generalizing previous results to broader classes of spectral functions.

Findings

Existence of solutions for all spectral indices k and dimensions d.

Solutions are bounded, connected, and have $C^{1,eta}$ boundaries outside a small singular set.

Results apply to a wide class of spectral functionals with bi-Lipschitz conditions.

Abstract

In this paper we prove that the shape optimization problem $$\min\left\{\lambda_k(\Omega):\ \Omega\subset\R^d,\ \Omega\ \hbox{open},\ P(\Omega)=1,\ |\Omega|<+\infty\right\},$$ has a solution for any $k\in\N$ 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.