[Regularity of interfaces for a Pucci type segregation problem

TL;DR

This paper proves the existence and regularity of interfaces in a two-phase free boundary problem involving Pucci-type operators, extending classical methods to fully nonlinear operators with different operators on each side.

Contribution

It introduces a novel approach to analyze regularity of free boundaries in fully nonlinear two-phase problems with different Pucci operators on each side.

Findings

Regular points on the free boundary are of class C^{1,α}.

The free boundary satisfies a weak form of the classical free boundary condition.

Provides an alternative proof for a segregation model with nonlinear diffusion.

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 $\partial \{ u> 0 \}\cap\Om$ 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 \cite{caffarelli_harnack_1987,caffarelli_harnack_1989}, and generalized by Wang \cite{wang_regularity_2000,wang_regularity_2002} and Feldman \cite{Fel} to fully nonlinear operators, to conclude that the regular points in $\partial \{ u> 0 \}\cap\Om$ form an open set of class $C^{1,\alpha}$. A novelty in our problem is that we have different operators, $\puccip$ and $\puccin$, on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators $\ppuccip$ and $\ppuccin$, our results provide an alternative approach to obtain the stationary limit %proof of existence to the one obtained from of a segregation model of populations with nonlinear diffusion in \cite{quitalo_free_2013}.