On a class of Cheeger inequalities

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.