On phase separation in systems of coupled elliptic equations: asymptotic analysis and geometric aspects

TL;DR

This paper investigates the asymptotic behavior of solutions to coupled elliptic systems with strong competition, providing sharp estimates near interfaces and characterizing the limit profiles as the competition parameter grows large.

Contribution

It offers new quantitative point-wise estimates and a regularity theory for interfaces in systems of coupled elliptic equations under high competition.

Findings

Sharp point-wise estimates near interfaces

Characterization of asymptotic profiles as competition diverges

Uniform regularity results for interfaces

Abstract

We consider a family of positive solutions to the system of $k$ components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in $\Omega$}, \] where $\Omega \subset \mathbb{R}^N$ with $N \ge 2$. It is known that uniform bounds in $L^\infty$ of $\{\mathbf{u}_{\beta}\}$ imply convergence of the densities to a segregated configuration, as the competition parameter $\beta$ diverges to $+\infty$. In this paper %we study more closely the asymptotic property of the solutions of the system in this singular limit: we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of $\mathbf{u}_\beta$ in terms of entire solutions to the limit system \[ \Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-$\beta$ regularity theory for the interfaces.