On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects
Nicola Soave, Alessandro Zilio · Annales de l Institut Henri Poincaré C Analyse Non Linéaire · 2016
We consider a family of positive solutions to the system of k components −\mathrm{\Delta }u_{i,\beta } = f(x,u_{i,\beta })−\beta u_{i,\beta }\sum \limits_{j eq i}a_{ij}u_{j,\beta }^{2}\:\text{in }\mathrm{\Omega }, where \mathrm{\Omega } \subset \mathbb{R}^{N} with N \geq 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 β diverges to +∞ . 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 \mathbf{u}_{\beta } in terms of entire solutions to the limit system \mathrm{\Delta }U_{i} = U_{i}\sum \limits_{j eq i}a_{ij}U_{j}^{2}. Moreover, we develop a uniform-in- β regularity theory for the interfaces.