On Two Species Long Range Segregation in an Annular Domain

Abstract: We consider a system of elliptic equations, depending on a small parameter $ε> 0$, which models the long range segregation of populations. The system has been previously studied in \cite{CL2} for the regularity of the free boundary in dimension $2$ and in \cite{ChPaTo26_2} and \cite{ChPaTo26_3} for the partial regularity of the free boundary in higher dimensions. In this paper, we consider the special case with $K = 2$ populations in an annular domain in arbitrary dimensions. Using the uniqueness and the rotational invariance of the solution, we show that the free boundary consist of concentric spheres. Moreover, by an application of the free boundary condition derived in \cite{CL2}, we show that the free boundary is uniquely determined. We also examine how the radius of the free boundary change according to the domain, the interaction distance and the boundary data. In particular, we show that the free boundary converges to that of the adjacent model as the interaction distance tends to zero. We also study the one parameter family of the elliptic system, in which the domain, the interaction distance, and the boundary data depends on $t$. We derive an ODE for the radius of the free boundary in dimension $2$ and dimension $n \geq 3$ separately. Several examples are given and discussed.