Published Paper
Inserted: 15 feb 2024
Last Updated: 14 jan 2025
Journal: J. Math. Pures Appl. (9)
Volume: 194
Pages: 103633
Year: 2025
Doi: 10.1016/j.matpur.2024.103633
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)).
Keywords: shape optimization, convex sets, Cheeger constant, cylinders, asymptotic estimates
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