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

We consider a family of positive solutions to the system of \(k\) components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in \(\Omega\)}, \] where \(\Omega \subset \mathbb{R}^N\) with \(N \ge 2\). It is known that uniform bounds in...

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Bibliographic Details
Published in:arXiv.org 2015-09
Main Authors: Soave, Nicola, Zilio, Alessandro
Format: Article
Language:English
Subjects:
Asymptotic properties
Elliptic functions
Phase separation
Online Access:Get full text
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Summary:We consider a family of positive solutions to the system of \(k\) components \[ -\Delta u_{i,\beta} = f(x, u_{i,\beta}) - \beta u_{i,\beta} \sum_{j \neq i} a_{ij} u_{j,\beta}^2 \qquad \text{in \(\Omega\)}, \] where \(\Omega \subset \mathbb{R}^N\) with \(N \ge 2\). It is known that uniform bounds in \(L^\infty\) of \(\{\mathbf{u}_{\beta}\}\) imply convergence of the densities to a segregated configuration, as the competition parameter \(\beta\) diverges to \(+\infty\). In this paper %we study more closely the asymptotic property of the solutions of the system in this singular limit: we establish sharp quantitative point-wise estimates for the densities around the interface between different components, and we characterize the asymptotic profile of \(\mathbf{u}_\beta\) in terms of entire solutions to the limit system \[ \Delta U_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-\(\beta\) regularity theory for the interfaces.
ISSN:2331-8422
DOI:10.48550/arxiv.1506.07779