The Bernoulli free boundary problem for the half Laplacian

In this talk I present some recent results of a joint work with Paolo Salani and Tadeusz Kulczycki concerning the fractional Bernoulli problem, where I will focus on the so called spectral half Laplacian. After a short introduction, I will explain how to construct a solution to this problem using the Beurling method on the extended problem. Moreover, we discuss geometric properties of the solution and, if there is time, give some remarks on the problem with the usual half Laplacian.