Uniqueness and least energy property for solutions to strongly competing systems (Q855162)

Summary: For the reaction-diffusion system of three competing species: \[ -\Delta u_i=-\mu u_i\sum_{j\neq i}u_j,\quad i=1,2,3, \] we prove uniqueness of the limiting configuration as \(\mu\to\infty\) on a planar domain \(\Omega\), with appropriate boundary conditions. Moreover, we prove that the limiting configuration minimizes the energy associated to the system \[ E(U)=\sum^3_{i=1}\int_\Omega\bigl| \nabla u_i(x) \bigr|^2dx \] among all segregated states \((u_i\cdot u_j=0\) a.e.) with the same boundary conditions.