On a class of Cheeger inequalities

TL;DR

This paper explores a generalized Cheeger inequality involving a shape functional based on eigenvalues, analyzing extremal properties and existence of optimal domains in both general and convex cases.

Contribution

It introduces a new shape functional $_{p,q}$ and studies its extremal values and the existence of optimal domains, extending classical Cheeger inequalities.

Findings

Characterization of infimum and supremum of $_{p,q}$ in all domains.

Existence results for optimal domains in the convex subclass.

Insights into the shape optimization problem related to eigenvalues.

Abstract

We study a general version of the Cheeger inequality by considering the shape functional $\mathcal{F}_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q(\Omega)^{1/q}$. The infimum and the supremum of $\mathcal{F}_{p,q}$ are studied in the class of all domains $\Omega$ of $\mathbb{R}^d$ and in the subclass of convex domains. In the latter case the issue concerning the existence of an optimal domain for $\mathcal{F}_{p,q}$ is discussed.