2015/05/20 by Caffarelli, Luis A., Quitalo, Veronica, Patrizi, Stefania · 2 citations
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1505.05433
In this work we study the properties of segregation processes modeled by a family of equations L(ui) (x) = ui(x) Fi (u1, …, uK)(x) i=1,…, K where Fi (u1, …, uK)(x) is a non-local factor that takes into consideration the values of the functions uj's in a full neighborhood of x. We consider as a model problem Δui^\ep (x) = \frac1\ep2 ui^\ep (x)∑i≠ j H(uj^\ep)(x) where \ep is a small parameter and H(uj^\ep)(x) is for instance H(uj^\ep)(x)= ∫B1 (x) uj^\ep (y) dy or H(uj^\ep)(x)= supy∈ B1(x) uj^\ep (y). Here the set B1(x) is the unit ball centered at x with respect to a smooth, uniformly convex norm ρ of \realn. Heuristically, this will force the populations to stay at ρ-distance 1, one from each other, as \ep→0.