On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects

Nicola Soave; Alessandro Zilio

doi:10.1016/j.anihpc.2016.04.001

JournalsaihpcVol. 34, No. 3pp. 625–654

On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects

Nicola Soave
Mathematisches Institut, Justus-Liebig-Universität Giessen, Arndtstrasse 2, 35392 Giessen, Germany
Alessandro Zilio
Centre d'Analyse et de Mathématique Sociales, École des Hautes Études en Sciences Sociales, 190-198 Avenue de France, 75244, Paris cedex 13, France

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Abstract

We consider a family of positive solutions to the system of $k$ components

- Δ u_{i, β} = f (x, u_{i, β}) - β u_{i, β} j \neq = i \sum a_{ij} u_{j, β}^{2} in Ω,

where $Ω \subset R^{N}$ with $N \geq 2$ . It is known that uniform bounds in $L^{\infty}$ of ${u_{β}}$ imply convergence of the densities to a segregated configuration, as the competition parameter $β$ diverges to $+ \infty$ . In this paper we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of $u_{β}$ in terms of entire solutions to the limit system

Δ U_{i} = U_{i} j \neq = i \sum a_{ij} U_{j}^{2} .

Moreover, we develop a uniform-in- $β$ regularity theory for the interfaces.

Cite this article

Nicola Soave, Alessandro Zilio, On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects. Ann. Inst. H. Poincaré Anal. Non Linéaire 34 (2017), no. 3, pp. 625–654

DOI 10.1016/J.ANIHPC.2016.04.001