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 \Omega ; with u and v being both positive in \Omega , vanishing simultaneously on \partial\Omega , and satisfying an overdetermined boundary value problem involving the product of their normal derivatives on \partial\Omega . Precisely, we consider solutions u, v \in C(B_1) of \Delta u= f \ \text{ and } \ -\Delta v=g \quad\text{in }\Omega=\{u>0\}=\{v>0\}, \frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q \quad\text{on }\partial\Omega\cap 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 \sqrt{uv} and \frac12(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,\alpha} regularity.