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CATEGORIES:Cambridge Centre for Analysis talks
SUMMARY:Asymptotics of stationary points of a non-local Gi
nzburg-Landau energy - Oxbridge PDE Days - Dr Dori
an Goldman\, DPMMS
DTSTART;TZID=Europe/London:20140320T150000
DTEND;TZID=Europe/London:20140320T160000
UID:TALK51577AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/51577
DESCRIPTION:We study a non-local Cahn-Hilliard energy arising
in the study of di-block copolymer melts\, often r
eferred to as the Ohta-Kawasaki energy in that con
text. In this model\, two phases appear\, which in
teract via a Coulombic energy. We focus on the reg
ime where one of the phases has a very small volum
e fraction\, thus creating ``droplets'' of the min
ority phase in a ``sea'' of the majority phase. In
this paper\, we address the asymptotic behavior o
f non-minimizing stationary points in dimensions $
n \\geq 2$ left open by the study of the $\\Gamma$
-convergence of the energy established in by Goldm
an-Muratov-Serfaty\, which provides information on
ly for almost minimizing sequences when $n=2$. In
particular\, we prove that (asymptotically) statio
nary points satisfy a force balance condition whic
h implies that the minority phase distributes itse
lf uniformly in the background majority phase. Our
proof uses and generalizes the framework of Sandi
er-Serfaty\, used in the context of stationary poi
nts of the Ginzburg-Landau model\, to higher dimen
sions. When $n=2$\, using the regularity results o
btained in\, we also are able to conclude that the
droplets in the sharp interface energy become asy
mptotically round when the number of droplets is c
onstrained to be finite and have bounded isoperime
tric deficit.\n\nhttp://www.maths.cam.ac.uk/postgr
ad/cca/oxbridgepde2014.html
LOCATION:MR2
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