Colloquium
3:00 p.m., Friday (November 19, 2004)
Math Annex 1100
Michael Ward
UBC
Eigenvalue Optimization, Spikes, and the Neumann Green's Function
An optimization problem for the fundamental eigenvalue of the
Laplacian in a planar simplyconnected domain that contains N
small identicallyshaped holes, each of a small radius \varepsilon\ll 1,
is considered. The boundary condition on the domain is assumed to be
of Neumann type, and a Dirichlet condition is imposed on the
boundary of each of the holes. The reciprocal of this eigenvalue is
proportional to the expected lifetime for Brownian motion in a
domain with a reflecting boundary that contains N small traps. For
small hole radii \varepsilon, we derive an asymptotic expansion for
this eigenvalue in terms of certain properties of the Neumann
Green's function for the Laplacian. This expansion depends on the
locations x_{i}, for i=1,\ldots,N, of the small holes. For the
unit disk, ringtype configurations of holes are constructed to optimize
the eigenvalue with respect to the hole locations. For arbitrary
symmetric dumbbellshaped domains containing exactly one hole, it is
shown that there is a unique hole location that maximizes the fundamental
eigenvalue. For an asymmetric dumbbellshaped domain, it is shown that
there can be two hole locations that locally maximize \lambda_0. This
eigenvalue optimization problem is shown to be closely related to determining
equilibrium locations of particlelike solutions, called spikes, to
certain singularly perturbed reactiondiffusion systems. Some interesting
properties of the equilibria, bifurcation behavior, and dynamics of these
particlelike solutions are discussed.
This is joint work with Theodore Kolokolnikov (UBC, Free University of Brussels),
and Michele Titcombe (CRM).
Refreshments will be served at 2:45 p.m. in the Faculty Lounge,
Math Annex (Room 1115).
