no. 3
On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects
Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 3, pp. 625-654.

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.

DOI: 10.1016/j.anihpc.2016.04.001
Classification: 35B45, 35B65, 35R35, 35B08, 35B36, 35B25, 35J47
Keywords: Nonlinear Schrödinger systems, Harmonic maps into singular manifolds, Competition and segregation, Point-wise asymptotic estimates, Regularity of free boundaries 
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     title = {On phase separation in systems of coupled elliptic equations: {Asymptotic} analysis and geometric aspects},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {625--654},
     publisher = {Elsevier},
     volume = {34},
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     year = {2017},
     doi = {10.1016/j.anihpc.2016.04.001},
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     zbl = {1417.35028},
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Soave, Nicola; Zilio, Alessandro. On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects. Annales de l'I.H.P. Analyse non linéaire, Volume 34 (2017) no. 3, pp. 625-654. doi : 10.1016/j.anihpc.2016.04.001. http://www.numdam.org/articles/10.1016/j.anihpc.2016.04.001/

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