Existence and regularity of optimal shapes for spectral functionals with Robin boundary conditions

TL;DR

This paper proves the existence and explores properties of optimal shapes minimizing spectral functionals with Robin boundary conditions, including results on eigenvalue equalities in higher dimensions.

Contribution

It establishes existence and qualitative properties of minimizers for spectral functionals with Robin boundary conditions, and confirms a conjecture about eigenvalue equalities in certain dimensions.

Findings

Existence of minimizers for spectral functionals with Robin boundary conditions.

Qualitative properties of these minimizers.

Validation of the eigenvalue equality conjecture in dimension three and higher.

Abstract

We establish the existence and find some qualitative properties of open sets that minimize functionals of the form $ F(\lambda_1(\Omega;\beta),\dots,\lambda_k(\Omega;\beta))$ under measure constraint on $\Omega$, where $\lambda_i(\Omega;\beta)$ designates the $i$-th eigenvalue of the Laplace operator on $\Omega$ with Robin boundary conditions of parameter $\beta>0$. Moreover, we show that minimizers of $\lambda_k(\Omega;\beta)$ for $k\geq 2$ verify the conjecture $\lambda_k(\Omega;\beta)=\lambda_{k-1}(\Omega;\beta)$ in dimension three and more.