Epsilon-regularity for the solutions of a free boundary system

This paper is dedicated to a free boundary system arising in the study of a class of shape optimization problems. The problem involves three variables: two functions u and v, anda domain st; with u and v being both positive in st, vanishing simultaneously on 8st, and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on 8st. Precisely, we consider solu-tions u, v E C(B1) of- u = f and -v = g in Omega = {u > 0} = {v > 0},partial derivative u partial derivative v/partial derivative n partial derivative n on partial derivative Omega boolean AND n B-1.Our main result is an epsilon-regularity theorem for viscosity solutions of this free boundary system. We prove a partial Harnack inequality near flat points for the couple of auxiliary functions ,/uv and 1/2 (u + v). Then, we use the gained space near the free boundary to transfer the improved flatness to the original solutions. Finally, using the partial Harnack inequality, we obtain an improvement-of-flatness result, which allows to conclude that flatness implies C-1,C-alpha regularity.