The fractional Cheeger problem

TL;DR

This paper studies the fractional Cheeger problem, a nonlocal optimization problem involving the ratio of fractional perimeter to volume, exploring properties of optimal sets and connections to eigenvalue problems.

Contribution

It introduces and analyzes the fractional Cheeger problem, providing new properties of optimal sets and linking the problem to nonlinear nonlocal eigenvalue limits.

Findings

Properties of optimal sets are established.

Equivalent formulations of the problem are provided.

Connections to eigenvalue problems are explored.

Abstract

Given an open and bounded set $\Omega\subset\mathbb{R}^N$, we consider the problem of minimizing the ratio between the $s-$perimeter and the $N-$dimensional Lebesgue measure among subsets of $\Omega$. This is the nonlocal version of the well-known Cheeger problem. We prove various properties of optimal sets for this problem, as well as some equivalent formulations. In addition, the limiting behaviour of some nonlinear and nonlocal eigenvalue problems is investigated, in relation with this optimization problem. The presentation is as self-contained as possible.