We show the existence of a Lipschitz viscosity solution u in Ω to a system of fully nonlinear equations involving Pucci-type operators. We study the regularity of the interface and we show that the viscosity inequalities of the system imply, in the weak sense, the free boundary condition , 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  to fully nonlinear operators, to conclude that the regular points in form an open set of class . A novelty in our problem is that we have different operators, and , on each side of the free boundary. In the particular case when these operators are the Pucci's extremal operators and , our results provide an alternative approach to obtain the stationary limit of a segregation model of populations with nonlinear diffusion in .
Cite this article
L. Caffarelli, S. Patrizi, V. Quitalo, M. Torres, Regularity of interfaces for a Pucci type segregation problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 36 (2019), no. 4, pp. 939–975DOI 10.1016/J.ANIHPC.2018.11.002