Uniqueness and least energy property for solutions to strongly competing systems

Monica Conti; Susanna Terracini; Gianmaria Verzini

doi:10.4171/ifb/150

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

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Abstract

For the reaction-diffusion system of three competing species:

- Δ u_{i} = - κ u_{i} j \neq = i \sum u_{j}, i = 1, 2, 3,

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

E (U) = i = 1 \sum 3 \int_{Ω} ∣\nabla u_{i} (x) ∣^{2} d x

among all segregated states ( $u_{i} \cdot u_{j} = 0$ 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