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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Energy driven pattern formation in a non-local Cah
n-Hilliard energy - Dorian Goldman (Cambridge)
DTSTART;TZID=Europe/London:20131021T150000
DTEND;TZID=Europe/London:20131021T160000
UID:TALK47575AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/47575
DESCRIPTION:This describes some joint work with Sylvia Serfaty
and Cyrill Muratov. We study \nthe asymptotic beh
avior of the screened sharp interface Ohta-Kawasak
i \nenergy in dimension 2. In that model\, \ntwo p
hases appear\, and they interact via a nonlocal Co
ulomb type energy. \nWe focus on the regime where
one of the phases has very small volume \nfraction
\, thus creating ``droplets" of that phase in a se
a of the other \nphase. We consider perturbations
to the critical volume fraction where \ndroplets f
irst appear\, show the number of droplets increase
s monotonically \nwith respect to the perturbation
factor\, and describe their arrangement in \nall
regimes\, whether their number is bounded or unbou
nded. When their \nnumber is unbounded\, the most
interesting case we compute the Γ limit of \nthe `
zeroth' order energy and yield averaged informatio
n for almost \nminimizers\, namely that the densit
y of droplets should be uniform. We then go to the
next order\, and derive a next order Γ-limit ener
gy\, which is exactly the ``Coulombian renormalize
d energy W" introduced in the work of Sandier/Serf
aty\, and obtained there as a limiting interaction
energy for vortices in Ginzburg-Landau. Without t
hus appealing at all to the Euler-Lagrange equatio
n\, we establish here for all configurations which
have ``almost minimal energy\," the asymptotic ro
undness and radius of the droplets as done by Mura
tov\, and the fact that they asymptotically shrink
to points whose arrangement should minimize the r
enormalized energy W\, in some averaged sense. Thi
s leads to expecting to see hexagonal lattices of
droplets. We also obtain analogous results for non
-minimizing critical points of the Ohta-Kawasaki e
nergy which hold in all dimensions\, and can concl
ude some information about the asymptotic roundnes
s of droplets in two dimensions.
LOCATION:CMS\, MR13
CONTACT:Prof. Clément Mouhot
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