The Free Boundary in a Higher-Dimensional Long-Range Segregation Model

TL;DR

This paper extends the analysis of free boundary regularity in a long-range population segregation model to higher dimensions, characterizing regular and singular points and their geometric properties.

Contribution

It generalizes the concept of angles and asymptotic cones to higher dimensions and provides a detailed structure of the free boundary, including conditions for regularity and convexity.

Findings

Regular set is a $C^1$ manifold of dimension $n-1$ under certain angle conditions.

Supports of populations are convex polytopes when convexity is assumed.

Characterization of regular and singular points based on densities and angles.

Abstract

We consider a system of elliptic equations, depending on a small parameter $\eps$, that models long-range segregation of populations. The diffusion is governed by the Laplacian. This system was previously investigated by Caffarelli, Patrizi, and Quitalo in \cite{CL2} as a model in population dynamics, and they established the regularity of the free boundary in two dimensions. In this paper we study the free boundary in the higher dimensional case. We extend the concept of angles and asymptotic cones to higher dimensions, and give a characterization of regular and singular points in terms of their densities and angles. We obtain a structure result of the free boundary and show that, if the angles at the singular points are away from $\frac{n\omega_n}{2}$, the regular set is open in the free boundary and locally a $C^1$ manifold of dimension $n-1$. We also show that, if the supports of the populations are convex, they are convex polytopes. A weak form of the equality of angles for the convex configuration is also derived.