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 |