A nonlinear model for long-range segregation

We study a system of fully nonlinear elliptic equations, depending on a small parameter , 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 {CL2} as a model in population dynamics. We establish the existence of solutions and prove convergence as →0⁺ 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.