Epsilon-regularity for the solutions of a free boundary system

Abstract

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.