Research interests:
Currently my research has
centered on questions
about how the solutions of
various elliptic
equations, or systems of
equations depend on
boundary conditions.
The simplest, and
prototypical, problem is
Laplace's equation. The
class of all solutions of
Laplace's equation on a
region G in space is
called the class of
harmonic functions on G. A
classical result is that,
when G is bounded and the
boundary of G is nice
enough, then the Dirichlet
solution map P is a
compact map from L^2(bdy
G) to
L^2(G). Such maps have a
singular value
decomposition (SVD)  an
analog (and
generalization) of the
spectral theorem for
compact linear maps of a
Hilbert space to itself.
In recent years I have
studied special bases of
Steklov eigenfunctions of
the Laplacian and the
biLaplacian that provide
orthogonal representations
and SVDs of the solutions
of Laplace's equations for
a number of different
types of boundary
conditions. With students,
we have found that these
provide simple and good
approximations of the
solutions of Laplace's
equations on some
different regions.
I am
particularly interested in
the application of these
ideas to various problems
in classical field
theories. Thus I am
studying issues about
approximations and
representations of
solutions of divcurl
systems, biharmonic
equations, the equations
of fluid mechanics and
electromagnetism.
Mathematically these
issues often are connected
with questions about trace
spaces and trace
inequalities. If you
have specific problems of
this type I am happy to
discuss what the possible
dependence of the
solutions on the boundary
data is and how such
dependence can be
quantified.
For a listing of recent
papers see Recent
Publications.
For a
full listing of research
papers, arranged by topic,
see Scientific
Publications.
See also Reviews
from MathSciNet.
