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 $$-\Delta u=f\text{and}-\Delta v=g\text{in}\Omega =\{u>0\}=\{v>0\},\frac{\partial u}{\partial n}\frac{\partial v}{\partial n}=Q\text{on}\partial \Omega \cap B_1.$$ 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 $\frac{1}{2}(u+v)$. 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 $C^{1,\alpha }$ regularity.