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CATEGORIES:Cambridge Analysts' Knowledge Exchange
SUMMARY:A result of existence and uniqueness for the Allen
-Cahn equation with singular potentials and dynami
c boundary conditions - Luca Calatroni\, (Cambridg
e Centre for Analysis)
DTSTART;TZID=Europe/London:20121114T160000
DTEND;TZID=Europe/London:20121114T170000
UID:TALK40604AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/40604
DESCRIPTION:In this talk we will present well-posedness result
s for the solution to an initial and boundary-valu
e problem for an Allen-Cahn type equation describi
ng the phenomenon of phase transition for a materi
al contained in a bounded and regular domain. The
nonlinearity appearing in the equation is frequent
ly assumed to be a double-well potential. In the t
alk we will generalize such nonlinearity consideri
ng a possibly non-smooth potential with domain dif
ferent from the whole real line. Such potentials a
re typically called _singular_ potentials. Even th
ough the Allen-Cahn equation is frequently coupled
with homogeneous Neumann boundary conditions (whi
ch are meant to represent the orthogonality of the
interface to the boundary and the absence of mass
flux)\, physicists have recently introduced the s
o-called _dynamic_ boundary conditions which take
into account the kinetics of the process on the bo
undary as well. In this talk we will consider simi
lar boundary conditions where another singular pot
ential appears. Both the uniqueness and the contin
uous dependence on data results hold in a quite ge
neral setting\, whereas for the proof of the exist
ence we need to enforce our assumptions\, introduc
ing a compatibility condition between the two sing
ular potentials. We prove the existence result by
regularizing the nonlinearities using Yosida appro
ximations and exploiting some _a priori_ estimates
that allow us to pass to the limit thanks to comp
actness and monotonicity results.\n\nThis is a joi
nt work with Prof. Pierluigi Colli\, University of
Pavia.
LOCATION:MR14\, Centre for Mathematical Sciences
CONTACT:Kevin Crooks
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