JournalsaihpcVol. 34, No. 6pp. 1387–1428

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

  • Serena Dipierro

    School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia

  • Enrico Valdinoci

    School of Mathematics and Statistics, University of Melbourne, 813 Swanston Street, Parkville, VIC 3010, Australia, Weierstraß Institut für Angewandte Analysis und Stochastik, Mohrenstraße 39, 10117 Berlin, Germany, Dipartimento di Matematica Federigo Enriques, Università degli Studi di Milano, Via Cesare Saldini 50, 20133 Milano, Italy, Istituto di Matematica Applicata e Tecnologie Informatiche Enrico Magenes, Consiglio Nazionale delle Ricerche, Via Ferrata 1, 27100 Pavia, Italy
Continuity and density results for a one-phase nonlocal free boundary problem cover
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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.

Cite this article

Serena Dipierro, Enrico Valdinoci, Continuity and density results for a one-phase nonlocal free boundary problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 6, pp. 1387–1428

DOI 10.1016/J.ANIHPC.2016.11.001