Optimal domains for the Cheeger inequality

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