Optimal Domains for the Cheeger Inequality

In this paper we consider the scale-invariant shape functional, where 1 ≤ q < p ≤ +∞ and λ p (Ω) (respectively λ q (Ω)) is the first eigenvalue of the p-Laplacian -∆ p (respectively -∆ q ) with Dirichlet boundary condition on ∂Ω.We study both the maximization and the minimization problems for F p,q , and show the existence of optimal domains in R d , along with some of their qualitative properties.Surprisingly, the case of a bounded box D constraint max λ q (Ω) : Ω ⊂ D, λ p (Ω) = 1 , leads to a problem of a different nature, for which the existence of a solution is shown through the analysis optimal capacitary measures.In the last section we list some interesting questions that, in our opinion, deserve to be investigated.