Calculus of Variations and Geometric Measure Theory

A. Pratelli - G. Saracco

Cylindrical estimates for the Cheeger constant and applications

created by saracco on 15 Feb 2024
modified on 16 Feb 2024


Submitted Paper

Inserted: 15 feb 2024
Last Updated: 16 feb 2024

Pages: 11
Year: 2024

ArXiv: 2402.09864 PDF


We prove a lower bound for the Cheeger constant of a cylinder $\Omega\times (0,L)$, where $\Omega$ 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 $\mathbb{R}^N$. This positively solves open conjectures raised by Parini (J. Convex Anal. (2017)) and by Briani–Buttazzo–Prinari (Ann. Mat. Pura Appl. (2023)).

Keywords: shape optimization, convex sets, Cheeger constant, cylinders, asymptotic estimates