Nonlocal phase transitions: Rigidity results and anisotropic geometry

We provide a series of rigidity results for a nonlocal phase transition equation. The prototype equation that we consider is of the form (-Δ)s/2u = u-u3, with s ϵ (0,1). More generally, we can take into account equations like Lu = f(u), where f is a bistable nonlinearity and L is an integro-differential operator, possibly of anisotropic type. The results that we obtain are an improvement of flatness theorem and a series of theorems concerning the one-dimensional symmetry for monotone and minimal solutions, in the research line dictated by a classical conjecture of E. De Giorgi. Here, we collect a series of pivotal results, of geometric type, which are exploited in the proofs of the main results.