A nonlinear model for long-range segregation

We study a system of fully nonlinear elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the negative Pucci operator. In the linear case, this system was previously investigated by Caffarelli, the second author, and Quitalo in \cite{CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as $\eps\to0^+$ to a free boundary problem in which populations remain segregated at a positive distance. In addition, we show that the supports of the limiting functions are sets of finite perimeter and satisfy a semi-convexity property.