Optimal domains for the Cheeger inequality

Abstract

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(\hat{I}\text{c◯})=\frac{\hat{I}{»}_p^{1/p}(\hat{I}\text{c◯})}{\hat{I}{»}_q^{1/q}(\hat{I}\text{c◯})},$$ where $1\le q<p\le +\infty$ and $\hat{I}{»}_p(\hat{I}\text{c◯})$ (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 $\partial \hat{I}\text{c◯}$. We study both the maximization and minimization problems for ${\mathcal{F}}_{p,q}$, and show the existence of optimal domains in ${\mathbb{R}}^d$, along with some of their qualitative properties. Surprisingly, the case of a bounded box $D$ constraint $$\max \{\hat{I}{»}_q(\hat{I}\text{c◯}):\hat{I}\text{c◯}\subset D,\hat{I}{»}_p(\hat{I}\text{c◯})=1\},$$ leads to a problem of different nature, for which the existence of a solution is shown by analyzing optimal capacitary measures. In the last section we list some interesting questions that, in our opinion, deserve to be investigated.

18 pages, 0 figures