Cylindrical estimates for the Cheeger constant and applications

Abstract

We prove a lower bound for the Cheeger constant of a cylinder $\Omega \times (0,L)$, where Ω is an open and bounded set. As a consequence, we obtain existence of minimizers for the shape functional defined as the ratio between the first Dirichlet eigenvalue of the p-Laplacian and the p-th power of the Cheeger constant, within the class of bounded convex sets in any $R^N$. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani–Buttazzo–Prinari (Ann. Mat. Pura Appl. (2023)).