The fractional Cheeger problem

Lorenzo Brasco (1) , Erik Lindgren (2) , Enea Parini (1)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 Department of Mathematical Sciences

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.

Mots clés: Cheeger constant; nonlocal eigenvalue problems; almost minimal surfaces

Domaines: Analyse fonctionnelle [math.FA]; Equations aux dérivées partielles [math.AP]

Lorenzo Brasco, Erik Lindgren, Enea Parini. The fractional Cheeger problem. Interfaces and Free Boundaries : Mathematical Analysis, Computation and Applications, 2014, 16, pp.419-458. ⟨10.4171/IFB/325⟩. ⟨hal-00864949v2⟩