Uniqueness and least energy property for solutions to strongly competing systems

  • Monica Conti

    Politecnico, Milano, Italy
  • Susanna Terracini

    Università di Torino, Italy
  • Gianmaria Verzini

    Politecnico di Milano, Italy

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 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 ( a.e.) with the same boundary conditions.

Cite this article

Monica Conti, Susanna Terracini, Gianmaria Verzini, Uniqueness and least energy property for solutions to strongly competing systems. Interfaces Free Bound. 8 (2006), no. 4, pp. 437–446

DOI 10.4171/IFB/150