Uniqueness and least energy property for solutions to strongly competing systems

Abstract

For the reaction--diffusion system of three competing species:

\[ -\D u_i=-\kappa u_i\sum_{j\neq i}u_j,\qquad i=1,2,3, \]

we prove uniqueness of the limiting configuration as $\kappa \to \infty$ on a planar domain \( \O \), with appropriate boundary conditions. Moreover we prove that the limiting configuration minimizes the energy associated to the system

\[ E(U)=\sum_{i=1}^3\int_\O|\nabla u_i(\bx)|^2\,d\bx \]

among all segregated states ($u_i\cdot u_j=0$ a.e.) with the same boundary conditions.