Gamma convergence of an energy functional related to the fractional Laplacian

We prove a Γ-convergence result for an energy functional related to some fractional powers of the Laplacian operator, (−Δ) ^s for 1/2 < s < 1, with two singular perturbations, that leads to a two-phase problem. The case (−Δ)^1/2 was considered by Alberti–Bouchitté–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.