Optimal domains for the Cheeger inequality

TL;DR

This paper establishes the existence of an optimal domain that maximizes a specific eigenvalue ratio involving the p-Laplacian, contributing to shape optimization and spectral theory.

Contribution

It proves the existence of an optimal domain for a shape optimization problem involving eigenvalues of the p-Laplacian, extending Cheeger inequality concepts.

Findings

Existence of an optimal domain for the eigenvalue ratio problem.

Connection between optimal domains and Cheeger ratio minimization.

Results applicable to shape optimization in spectral theory.

Abstract

In this paper we prove the existence of an optimal domain $\Omega_{opt}$ for the shape optimization problem $$\max\Big\{\lambda_q(\Omega)\ :\ \Omega\subset D,\ \lambda_p(\Omega)=1\Big\},$$ where $q<p$ and $D$ is a prescribed bounded subset of ${\bf R}^d$. Here $\lambda_p(\Omega)$ (respectively $\lambda_q(\Omega)$) is the first eigenvalue of the $p$-Laplacian $-\Delta_p$ (respectively $-\Delta_q$) with Dirichlet boundary condition on $\partial\Omega$. This is related to the existence of optimal sets that minimize the generalized Cheeger ratio $${\mathcal F}_{p,q}(\Omega)=\frac{\lambda_p^{1/p}(\Omega)}{\lambda_q^{1/q}(\Omega)}.$$