WIAS Preprint No. 2101, (2015)

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


  • Dipierro, Serena
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 35R35 49N60 35R11, 35A15


  • Free boundary problems, nonlocal minimalsurfaces, fractional operators, regularity theory, fractional harmonic replacement




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 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.

Appeared in

  • Ann. Inst. H. Poincare Anal. Non Lineaire, 34 (2017), pp. 1387-1428.

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