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 $$-\mathrm{\Delta }{u}_{i,\beta }=f(x,u_{i,\beta })-\beta u_{i,\beta }\sum _{j\mathrm{\ne }i}{a}_{ij}{u}_{j,\beta }^2\text{in }\mathrm{\Omega },$$ where $\Omega \subset $\mathbb{R}$^N$with$N \ge 2$\mathrm{.}Itisknownthatuniformboundsin$L^\infty$of${$\mathbf{u}$_{\beta}}$ imply convergence of the densities to a segregated configuration, as the competition parameter $\beta$ diverges to $+\mathrm{\infty }$. 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 $$\mathrm{\Delta }{U}_i=U_i\sum _{j\mathrm{\ne }i}{a}_{ij}{U}_j^2\mathrm{.}$$ Moreover, we develop a uniform-in-$\beta$ regularity theory for the interfaces.