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 Hölder 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.
|Number of pages||42|
|Journal||Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire|
|Publication status||Published - 2017|