A nonlocal free boundary problem

TL;DR

This paper studies a nonlocal free boundary problem involving fractional Sobolev seminorms and fractional perimeter, establishing key properties like monotonicity, regularity, and convergence, with connections to classical free boundary problems.

Contribution

It introduces a new nonlocal free boundary problem framework and proves foundational results including monotonicity, regularity, and limit case analyses.

Findings

Proved a monotonicity formula for minimizers.

Established regularity results for planar cones.

Analyzed limit cases connecting to classical free boundary problems.

Abstract

Given~$s,\sigma\in(0,1)$ and a bounded domain~$\Omega\subset\R^n$, we consider the following minimization problem of $s$-Dirichlet plus $\sigma$-perimeter type $$ [u]_{ H^s(\R^{2n}\setminus(\Omega^c)^2) } + \Per_\sigma\left(\{u>0\},\Omega\right), $$ where~$[ \cdot]_{H^s}$ is the fractional Gagliardo seminorm and $\Per_\sigma$ is the fractional perimeter. Among other results, we prove a monotonicity formula for the minimizers, glueing lemmata, uniform energy bounds, convergence results, a regularity theory for the planar cones and a trivialization result for the flat case. Several classical free boundary problems are limit cases of the one that we consider in this paper, as $s\nearrow1$, $\sigma\nearrow1$ or~$\sigma\searrow0$.