Regularity of the free boundary in two-phase problems for linear elliptic equations

In this paper we complete the study of the regularity of the free boundary in two-phase problems for linear elliptic operators started in [M.C. Cerutti, F. Ferrari, S. Salsa, Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are C1,γ , Arch. Ration. Mech. Anal. 171 (2004) 329–348]. In particular we prove that Lipschitz and flat free boundaries (in a suitable sense) are smooth. As byproduct, we prove that Lipschitz free boundaries are smooth in the case of quasilinear operators of the form div(A(x, u)∇u) with Lipschitz coefficients.