Optimal domains for the Cheeger inequality

Abstract

In this paper we prove the existence of an optimal domain $\hat{I}{\text{c◯}}_{opt}$ for the shape optimization problem $$\max \{\hat{I}{»}_q(\hat{I}\text{c◯}):\hat{I}\text{c◯}\subset D,\hat{I}{»}_p(\hat{I}\text{c◯})=1\},$$ where $q<p$ and $D$ is a prescribed bounded subset of ${\mathbf{R}}^d$. Here $\hat{I}{»}_p(\hat{I}\text{c◯})$ (respectively $\hat{I}{»}_q(\hat{I}\text{c◯})$) is the first eigenvalue of the $p$-Laplacian $-\hat{I}{”}_p$ (respectively $-\hat{I}{”}_q$) with Dirichlet boundary condition on $\partial \hat{I}\text{c◯}$. This is related to the existence of optimal sets that minimize the generalized Cheeger ratio $${\mathcal{F}}_{p,q}(\hat{I}\text{c◯})=\frac{\hat{I}{»}_p^{1/p}(\hat{I}\text{c◯})}{\hat{I}{»}_q^{1/q}(\hat{I}\text{c◯})}.$$

there is an improved version in preparation