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SUMMARY:Energy driven pattern formation in a non-local Ginzburg-Landau/Cah
 n-Hilliard energy - Dorian Goldman (NYU)
DTSTART:20121122T150000Z
DTEND:20121122T160000Z
UID:TALK41645@talks.cam.ac.uk
CONTACT:Prof. Mihalis Dafermos
DESCRIPTION:This describes joint work with Sylvia Serfaty and Cyrill Murat
 ov. We study the asymptotic behavior of the screened sharp interface Ohta-
 Kawasaki energy in dimension 2 using the framework of \\Gamma-convergence.
  In that model\, two phases appear\, and they interact via a nonlocal Coul
 omb type energy. We focus on the regime where one of the phases has very s
 mall volume fraction\, thus creating "droplets" of that phase in a sea of 
 the other phase. We consider perturbations to the critical volume fraction
  where droplets first appear\, show the number of droplets increases monot
 onically with respect to the perturbation factor\, and describe their arra
 ngement in all regimes\, whether their number is bounded or unbounded. Whe
 n their number is unbounded\, the most interesting case we compute the \\G
 amma limit of the "zeroth" order energy and yield averaged information for
  almost minimizers\, namely that the density of droplets should be uniform
 . We then go to the next order\, and derive a next order \\Gamma-limit ene
 rgy\, which is exactly the "Coulombian renormalized energy W" introduced i
 n the work of Sandier/Serfaty\, and obtained there as a limiting interacti
 on energy for vortices in Ginzburg-Landau. The derivation is based on thei
 r abstract scheme\, that serves to obtain lower bounds for 2-scale energie
 s and express them through some probabilities on patterns via the multipar
 ameter ergodic theorem. Without thus appealing at all to the Euler-Lagrang
 e equation\, we establish here for all configurations which have "almost m
 inimal energy\," the asymptotic roundness and radius of the droplets as do
 ne by Muratov\, and the fact that they asymptotically shrink to points who
 se arrangement should minimize the renormalized energy W\, in some average
 d sense. This leads to expecting to see hexagonal lattices of droplets.
LOCATION:CMS\, MR3
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