Two-phase problems for linear elliptic operators with variable coefficients: Lipschitz free boundaries are $C^{1,\gamma}$.

This paper proves regularity of Lipschitz free boundaries of a two-phase problem for linear elliptic operators. The main strategy of proof involves a nontrivial extension of the work of L. A. Caffarelli in [Rev. Mat. Iberoamericana 3 (1987), no. 2, 139--162; MR0990856 (90d:35306)] and [Comm. Pure Appl. Math. 42 (1989), no. 1, 55--78; MR0973745 (90b:35246)]. It is carried out through several stages of perturbation analysis around the results in the constant coefficient case