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, and a domain Ω; with u and v being both positive in Ω, vanishing simultaneously on @Ω, and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on @Ω. Precisely, we consider solutions u; v 2 C.B1/ of 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 puv and 21.u C 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 C1,αregularity. (Formula Presented)