Optimization of the anisotropic Cheeger constant with respect to the anisotropy

TL;DR

This paper investigates how to optimize the anisotropic Cheeger constant by adjusting the anisotropy shape under volume constraints, proving existence of minimizers and identifying optimal shapes in specific cases.

Contribution

It establishes the existence of minimizers for the anisotropic Cheeger constant in the planar case and characterizes optimal anisotropies for specific domains.

Findings

Existence of minimizers for the anisotropic Cheeger constant in the planar case.

Optimal anisotropy for a ball is not a ball, but a square provides the minimal value among polygons.

The study provides insights into shape optimization under anisotropic conditions.

Abstract

Given an open, bounded set $\Omega$ in $\mathbb{R}^N$, we consider the minimization of the anisotropic Cheeger constant $h_K(\Omega)$ with respect to the anisotropy $K$, under a volume constraint on the associated unit ball. In the planar case, under the assumption that $K$ is a convex, centrally symmetric body, we prove the existence of a minimizer. Moreover, if $\Omega$ is a ball, we show that the optimal anisotropy $K$ is not a ball and that, among all regular polygons, the square provides the minimal value.