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CATEGORIES:Partial Differential Equations seminar
SUMMARY:Energy driven pattern formation in a non-local Gin
zburg-Landau/Cahn-Hilliard energy - Dorian Goldman
(NYU)
DTSTART;TZID=Europe/London:20121122T150000
DTEND;TZID=Europe/London:20121122T160000
UID:TALK41645AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/41645
DESCRIPTION:This describes joint work with Sylvia Serfaty and
Cyrill Muratov. We study the asymptotic behavior o
f the screened sharp interface Ohta-Kawasaki energ
y in dimension 2 using the framework of \\Gamma-co
nvergence. In that model\, two phases appear\, and
they interact via a nonlocal Coulomb type energy.
We focus on the regime where one of the phases ha
s very small volume fraction\, thus creating "drop
lets" of that phase in a sea of the other phase. W
e consider perturbations to the critical volume fr
action where droplets first appear\, show the numb
er of droplets increases monotonically with respec
t to the perturbation factor\, and describe their
arrangement in all regimes\, whether their number
is bounded or unbounded. When their number is unbo
unded\, the most interesting case we compute the \
\Gamma limit of the "zeroth" order energy and yiel
d averaged information for almost minimizers\, nam
ely that the density of droplets should be uniform
. We then go to the next order\, and derive a next
order \\Gamma-limit energy\, which is exactly the
"Coulombian renormalized energy W" introduced in
the work of Sandier/Serfaty\, and obtained there a
s a limiting interaction energy for vortices in Gi
nzburg-Landau. The derivation is based on their ab
stract scheme\, that serves to obtain lower bounds
for 2-scale energies and express them through som
e probabilities on patterns via the multiparameter
ergodic theorem. Without thus appealing at all to
the Euler-Lagrange equation\, we establish here f
or all configurations which have "almost minimal e
nergy\," the asymptotic roundness and radius of th
e droplets as done by Muratov\, and the fact that
they asymptotically shrink to points whose arrange
ment should minimize the renormalized energy W\, i
n some averaged sense. This leads to expecting to
see hexagonal lattices of droplets.
LOCATION:CMS\, MR3
CONTACT:Prof. Mihalis Dafermos
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