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

Soave, Nicola; Zilio, Alessandro

doi:10.1016/j.anihpc.2016.04.001

Soave, Nicola ; Zilio, Alessandro

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 3, pp. 625-654

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Résumé

We consider a family of positive solutions to the system of k components

- Δ u_{i, β} = f (x, u_{i, β}) - β u_{i, β} \sum_{j \neq i} a_{i j} u_{j, β}^{2} in Ω,

where

Ω \subset R^{N}

with

N \geq 2

. It is known that uniform bounds in

L^{\infty}

of

{u_{β}}

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

u_{β}

in terms of entire solutions to the limit system

Δ U_{i} = U_{i} \sum_{j \neq i} a_{i j} U_{j}^{2} .

Moreover, we develop a uniform-in-β regularity theory for the interfaces.

MR Zbl

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

@article{AIHPC_2017__34_3_625_0,
     author = {Soave, Nicola and Zilio, Alessandro},
     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},
     year = {2017},
     publisher = {Elsevier},
     volume = {34},
     number = {3},
     doi = {10.1016/j.anihpc.2016.04.001},
     mrnumber = {3633738},
     zbl = {1417.35028},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1016/j.anihpc.2016.04.001/}
}

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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, Tome 34 (2017) no. 3, pp. 625-654. doi: 10.1016/j.anihpc.2016.04.001

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