Gamma convergence of an energy functional related to the fractional Laplacian

We prove a I-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (-Delta) (s) for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (-Delta)(1/2) was considered by Alberti-Bouchitt,-Seppecher in relation to a model in capillarity with line tension effect. However, the proof in our setting requires some new ingredients such as the Caffarelli-Silvestre extension for the fractional Laplacian and new trace inequalities for weighted Sobolev spaces.