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arXiv · 2402.09864

Cylindrical estimates for the Cheeger constant and applications

Aldo Pratelli·Giorgio Saracco
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Abstract

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)).

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Aldo Pratelli, Giorgio Saracco. 2024-02-15. Cylindrical estimates for the Cheeger constant and applications. https://doi.org/10.1016/j.matpur.2024.103633

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