Geodesic
Continuity and density results for a one-phase nonlocal free boundary problem
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1387-1428
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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.

DOI : 10.1016/j.anihpc.2016.11.001
Classification : 35R35, 49N60, 35R11, 35A15
Keywords: Free boundary problems, Nonlocal minimal surfaces, Fractional operators, Regularity theory, Fractional harmonic replacement 
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     author = {Dipierro, Serena and Valdinoci, Enrico},
     title = {Continuity and density results for a one-phase nonlocal free boundary problem},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1387--1428},
     year = {2017},
     publisher = {Elsevier},
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     number = {6},
     doi = {10.1016/j.anihpc.2016.11.001},
     zbl = {1386.35490},
     mrnumber = {3712006},
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     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.11.001/}
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Dipierro, Serena; Valdinoci, Enrico. Continuity and density results for a one-phase nonlocal free boundary problem. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 6, pp. 1387-1428. doi: 10.1016/j.anihpc.2016.11.001

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