Optimal domains for the Cheeger inequality

In this paper we consider the scale invariant shape functional $${\mathcal{F}}_{p,q}(Ω)=\frac{λ_p^{1/p}(Ω)}{λ_q^{1/q}(Ω)},$$ where $1\le q<p\le+\infty$ and $λ_p(Ω)$ (respectively $λ_q(Ω)$) is the first eigenvalue of the $p$-Laplacian $-Δ_p$ (respectively $-Δ_q$) with Dirichlet boundary condition on $\partialΩ$. 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 $$\max\Big\{λ_q(Ω)\ :\ Ω\subset D,\ λ_p(Ω)=1\Big\},$$ 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.