Regularity in a one-phase free boundary problem for the fractional Laplacian

D. De Silva; J.M. Roquejoffre

doi:10.1016/j.anihpc.2011.11.003

JournalsaihpcVol. 29, No. 3pp. 335–367

Regularity in a one-phase free boundary problem for the fractional Laplacian

D. De Silva
Department of Mathematics, Barnard College, Columbia University, New York, NY 10027, United States
J.M. Roquejoffre
Institut de Mathematiques (UMR CNRS 5640), Université Paul Sabatier, 31062 Toulouse Cedex 4, France

$Regularity in a one-phase free boundary problem for the fractional Laplacian cover$

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Abstract

For a one-phase free boundary problem involving a fractional Laplacian, we prove that “flat free boundaries” are $C^{1, α}$ . We recover the regularity results of Caffarelli for viscosity solutions of the classical Bernoulli-type free boundary problem with the standard Laplacian.

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D. De Silva, J.M. Roquejoffre, Regularity in a one-phase free boundary problem for the fractional Laplacian. Ann. Inst. H. Poincaré Anal. Non Linéaire 29 (2012), no. 3, pp. 335–367

DOI 10.1016/J.ANIHPC.2011.11.003