Regularity of interfaces for a Pucci type segregation problem

Abstract

We show the existence of a Lipschitz viscosity solution $u$ in $\Omega$ to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface $\partial \{u>0\}\cap \mathrm{\Omega }$ and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition $u_{{\nu }_{+}}^{+}=u_{{\nu }_{-}}^{-}$, and hence $u$ is a solution to a two-phase free boundary problem. We show that we can apply the classical method of sup-convolutions developed by the first author in [5,6], and generalized by Wang [20,21] and Feldman [11] to fully nonlinear operators, to conclude that the regular points in $\partial \{u>0\}\cap \mathrm{\Omega }$ form an open set of class $C^{1,\alpha }$. A novelty in our problem is that we have different operators, ${\mathcal{F}}^{+}$ and ${\mathcal{F}}^{-}$, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators ${\mathcal{M}}^{+}$ and ${\mathcal{M}}^{-}$, our results provide an alternative approach to obtain the stationary limit of a segregation model of populations with nonlinear diffusion in [19].