A free boundary problem arising from segregation of populations with high competition

TL;DR

This paper derives a free boundary problem from a nonlinear elliptic system modeling population segregation with high competition, analyzing solution regularity and Lipschitz continuity across the free boundary.

Contribution

It introduces a new free boundary problem as a limit of a nonlinear elliptic system for population segregation and proves Lipschitz regularity of solutions.

Findings

Established the free boundary problem as a limit of the nonlinear system.

Proved Lipschitz regularity of solutions across the free boundary.

Extended regularity results to a nonlinear, non-smooth setting.

Abstract

In this work, we show how to obtain a free boundary problem as the limit of a fully non linear elliptic system of equations that models population segregation (Gause-Lotka-Volterra type). We study the regularity of the solutions. In particular, we prove Lipschitz regularity across the free boundary. The problem is motivated by the work done by Caffarelli, Karakhanyan and Fang-Hua Lin for the linear case.