Thursday, October 23, 2025 — 2:30 pm — aula M1.4, edificio Matematica
Optimal domains for the Cheeger inequality
Abstract: We study a generalized form of the Cheeger inequality by considering the shape functional $$F_{p,q}(\Omega)=\lambda_p^{1/p}(\Omega)/\lambda_q^{1/q}(\Omega) \: ,$$ where the original Cheeger case corresponds to $p=2$ and $q=1$. Here $\lambda_p(\Omega)$ denotes the principal eigenvalue of the Dirichlet $p$-Laplacian. The infimum and the supremum of $F_{p,q}$ are discussed, together with the existence of optimal domains. Some open problems will be illustrated as well.