Optimal domains for the Cheeger inequality

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