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SUMMARY:Energy driven pattern formation in a non-local Cahn-Hilliard energ
 y - Dorian Goldman (Cambridge)
DTSTART:20131021T140000Z
DTEND:20131021T150000Z
UID:TALK47575@talks.cam.ac.uk
CONTACT:Prof. Clément Mouhot
DESCRIPTION:This describes some joint work with Sylvia Serfaty and Cyrill 
 Muratov. We study \nthe asymptotic behavior of the screened sharp interfac
 e Ohta-Kawasaki \nenergy in dimension 2. In that model\, \ntwo phases appe
 ar\, and they interact via a nonlocal Coulomb type energy. \nWe focus on t
 he regime where one of the phases has very small volume \nfraction\, thus 
 creating ``droplets" of that phase in a sea of the other \nphase. We consi
 der perturbations to the critical volume fraction where \ndroplets first a
 ppear\, show the number of droplets increases monotonically \nwith respect
  to the perturbation factor\, and describe their arrangement in \nall regi
 mes\, whether their number is bounded or unbounded. When their \nnumber is
  unbounded\, the most interesting case we compute the Γ limit of \nthe `z
 eroth' order energy and yield averaged information for almost \nminimizers
 \, namely that the density of droplets should be uniform. We then go to th
 e next order\, and derive a next order Γ-limit energy\, which is exactly 
 the ``Coulombian renormalized energy W" introduced in the work of Sandier/
 Serfaty\, and obtained there as a limiting interaction energy for vortices
  in Ginzburg-Landau. Without thus appealing at all to the Euler-Lagrange e
 quation\, we establish here for all configurations which have ``almost min
 imal energy\," the asymptotic roundness and radius of the droplets as done
  by Muratov\, and the fact that they asymptotically shrink to points whose
  arrangement should minimize the renormalized energy W\, in some averaged 
 sense. This leads to expecting to see hexagonal lattices of droplets. We a
 lso obtain analogous results for non-minimizing critical points of the Oht
 a-Kawasaki energy which hold in all dimensions\, and can conclude some inf
 ormation about the asymptotic roundness of droplets in two dimensions.
LOCATION:CMS\, MR13
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