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

Abstract

We consider a family of positive solutions to the system of $k$ components \[ -Δu_{i,β} = f(x, u_{i,β}) - βu_{i,β} \sum_{j \neq i} a_{ij} u_{j,β}^2 \qquad \text{in $\Omega$}, \] 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 \[ ΔU_i = U_i \sum_{j\neq i} a_{ij} U_j^2. \] Moreover, we develop a uniform-in-$\beta$ regularity theory for the interfaces.

33 pages, some correction with respect to the previous version

References in corpus (4)

• Uniform bounds for strongly competing systems: the optimal Lipschitz case
• Liouville theorems and $1$-dimensional symmetry for solutions of an elliptic system modelling phase separation
• Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues
• Multidimensional entire solutions for an elliptic system modelling phase separation

Cited by in corpus (6)

• Variational problems with long-range interaction
• Multidimensional entire solutions for an elliptic system modelling phase separation
• On the Asymptotic Growth of Positive Solutions to a Nonlocal Elliptic Blow-up System Involving Strong Competition
• An anisotropic monotoncity formula, with applications to some segregation problems
• Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem
• On the solvability of a linear inhomogeneous problem arising in the blow-up analysis of the phase separation in Bose-Einstein condensates