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.

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

SOAVE, NICOLA;
2017

Abstract

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.
Competition and segregation; Harmonic maps into singular manifolds; Nonlinear Schrödinger systems; Point-wise asymptotic estimates; Regularity of free boundaries; Analysis; Mathematical Physics
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11311/1028261
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