On a long range segregation model

Luis A. Caffarelli; Stefania Patrizi; Veronica Quitalo

doi:10.4171/jems/747

JournalsjemsVol. 19, No. 12pp. 3575–3628

On a long range segregation model

Luis A. Caffarelli
University of Texas at Austin, USA
Stefania Patrizi
University of Texas at Austin, USA
Veronica Quitalo
Purdue University, West Lafayette, USA

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Abstract

In this work we study the properties of segregation processes modeled by a family of equations

L (u_{i}) (x) = u_{i} (x) F_{i} (u_{1}, \dots, u_{K}) (x) i = 1, \dots, K

where $F_{i} (u_{1}, \dots, u_{K}) (x)$ is a non-local factor that takes into consideration the values of the functions $u_{j}$ in a full neighborhood of $x .$ We consider as a model problem

Δ u_{i}^{ϵ} (x) = \frac{1}{ϵ ^{2}} u_{i}^{ϵ} (x) i \neq = j \sum H (u_{j}^{ϵ}) (x)

where $ϵ$ is a small parameter and $H (u_{j}^{ϵ}) (x)$ is for instance

H (u_{j}^{ϵ}) (x) = \int_{B_{1} (x)} u_{j}^{ϵ} (y) d y or H (u_{j}^{ϵ}) (x) = sup_{y \in B_{1} (x)} u_{j}^{ϵ} (y) .

Here $B_{1} (x)$ is the unit ball centered at $x$ with respect to a smooth, uniformly convex norm $ρ$ of $R^{n}$ . Heuristically, this will force the populations to stay at $ρ$ -distance 1, one from each other, as $ϵ \to 0$ .

Cite this article

Luis A. Caffarelli, Stefania Patrizi, Veronica Quitalo, On a long range segregation model. J. Eur. Math. Soc. 19 (2017), no. 12, pp. 3575–3628

DOI 10.4171/JEMS/747