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A one-phase problem for the fractional Laplacian: regularity of flat free boundaries. (English) Zbl 1290.35342

Summary: We consider a one-phase free boundary problem involving a fractional Laplacian \((-\Delta)^{\alpha}\), \(0<\alpha<1\), and prove that “flat free boundaries” are \(C^{1,\gamma}\). We thus extend the known results for the case \(\alpha=1/2\).

MSC:

35R35 Free boundary problems for PDEs
35R11 Fractional partial differential equations
35B09 Positive solutions to PDEs
35B50 Maximum principles in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35J60 Nonlinear elliptic equations