Flat free boundaries regularity in two-phase problems for a class of fully nonlinear elliptic operators with variable coefficients

Abstract

In this paper we prove that flat free boundaries of the solutions of elliptic two-phase problems associated with a class of fully nonlinear operators are C1,γ . In [11] the C1,γ regularity of Lipschitz free boundaries of two-phase problems was proved for a class of homogeneous fully nonlinear elliptic operators F(D2u(x), x), containing convex (concave) operators, with Holder dependence on x. Here we consider the same class of operators. More precisely, we prove the regularity of flat free boundaries of the solutions of the following two-phase problems:  F(D2u(x), x) = 0 in Ω(u) = {x ∈ Ω ⊂ R : u > 0}, F (D2u(x), x) = 0 in Ω(u) = {x ∈ Ω ⊂ R : u 6 0}, u = 0 on Fu, u+ν = G(u − ν ) on Fu, (1)