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 −Δui,β=f(x,ui,β)−βui,β∑j≠iaijuj,β2in Ω, where Ω⊂RN with N≥2. It is known that uniform bounds in L∞ of uβ 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 uβ in terms of entire solutions to the limit system ΔUi=Ui∑j≠iaijUj2. Moreover, we develop a uniform-in-β regularity theory for the interfaces.