Calculus of Variations and Geometric Measure Theory

L. Brasco - E. Lindgren - E. Parini

The fractional Cheeger problem

created by brasco on 19 Aug 2013
modified on 07 May 2014


Accepted Paper

Inserted: 19 aug 2013
Last Updated: 7 may 2014

Journal: Interfaces Free Bound.
Pages: 34
Year: 2013


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 {\it 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.

Keywords: Cheeger constant, nonlocal eigenvalue problems, almost minimal surfaces