We discuss properties (optimal regularity, non-degeneracy, smoothness of the free boundary...) of a variational interface problem involving the fractional Laplacian; Due to the non-locality of the Dirichlet problem, the task is nontrivial. This difficulty is by-passed by an extension formula, discovered by the first author and Silvestre, which reduces the study to that of a co-dimension 2 (degenerate) free boundary.
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Luis A. Caffarelli, Jean-Michel Roquejoffre, Yannick Sire, Variational problems with free boundaries for the fractional Laplacian. J. Eur. Math. Soc. 12 (2010), no. 5, pp. 1151–1179DOI 10.4171/JEMS/226