Modelling phase separation with coupled elliptic equations: recent results on the asymptotic analysis

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Date/heure
Date(s) - 01/03/2016
11 h 00 min - 12 h 00 min

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We consider a family of positive solutions to the system of k components $-\Delta u_{i,\beta}=f(x,u_{i,\beta})-\beta u_{i,\beta}\sum_{j\neq i}a_{i,j}u_{j,\beta}^2$ in $\Omega \subset R^N$ with $N\geq 2$. It is known that uniform bounds in $L^{\infty}$ of $\lbrace u_{\beta}\rbrace$ imply convergence of the densities to a segregated configuration, as the competition parameter $\beta$ diverges to $+\infty$. In this talk I will discuss how to obtain sharp quantitative point-wise estimates for the densities around the interface between different components, and, more specifically, how to characterize the asymptotic profile of $u_{\beta}$ in terms of entire solutions to the limit system $-\Delta U_i = U_i\sum_{j\neq i}a_{ij}U_j^2$. These results are part of an ongoing project with Nicola Soave.

Olivier CHABROL
Posts created 14

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