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.
Continuity and density results for a one-phase nonlocal free boundary problem / S. Dipierro, E. Valdinoci. - In: ANNALES DE L INSTITUT HENRI POINCARÉ. ANALYSE NON LINÉAIRE. - ISSN 0294-1449. - 34:6(2017), pp. 1387-1428.
Continuity and density results for a one-phase nonlocal free boundary problem
S. Dipierro;E. Valdinoci
2017
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 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.
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