Continuity and density results for a one-phase nonlocal free boundary problem

TL;DR

This paper studies a nonlocal free boundary problem combining fractional energy and perimeter, proving regularity of solutions and positivity of the free boundary density, and introduces a new fractional harmonic replacement concept.

Contribution

It introduces a novel fractional harmonic replacement in extended variables and establishes regularity and density properties for minimizers of the nonlocal free boundary problem.

Findings

Minimizers are Hölder continuous.

The free boundary has positive density from both sides.

A new fractional harmonic replacement concept is developed.

Abstract

We consider a one-phase nonlocal free boundary problem obtained by the superposition of a fractional Dirichlet energy plus a nonlocal perimeter functional. We prove that the minimizers are H\"older continuous and the free boundary has positive density from both sides. For this, we also introduce a new notion of fractional harmonic replacement in the extended variables and we study its basic properties.