JournalsifbVol. 8, No. 4pp. 437–446

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
Uniqueness and least energy property for solutions to strongly competing systems cover
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Abstract

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

\Dui=κuijiuj,i=1,2,3,-\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 κ\kappa\to\infty on a planar domain \O\O, with appropriate boundary conditions. Moreover we prove that the limiting configuration minimizes the energy associated to the system

E(U)=i=13\Oui(\bx)2d\bxE(U)=\sum_{i=1}^3\int_\O|\nabla u_i(\bx)|^2\,d\bx

among all segregated states (uiuj=0u_i\cdot u_j=0 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