Speaker: Xiaofeng Ren
(George Washington University)
Title: Analysis and computation of some pattern formation problems
Abstract: The Ohta-Kawasaki density functional theory of
block
copolymers is a variational problem with a nonlocal term in the
integrand. It models the lamellar, cylindrical, and spherical
morphological phases observed in diblock copolymers.
The Gierer-Meinhardt problem is a reaction-diffusion system of
partial
differential equations for pattern formation and morphogenesis
observed
in the development of a higher organism out of a single
fertilized egg. It is a minimal model that provides a theoretical
bridge
between observations on the one hand and the deduction of the
underlying
molecular-genetic mechanisms on the other hand.
Interestingly in some important parameter ranges, these two
different
problems converge to the same singular limit: a geometric problem
of
finding a subset in plane or space that satisfies an equation
involving
the curvature of the boundary of the set.
In this talk, I will present some direct computations that
numerically
simulate the Ohta-Kawasaki theory. Then I will discuss some
analytic
result that further reduce the geometric singular limit to a
finite
dimensional problem. And finally the finite dimensional problem is
solved numerically by Matlab.