Research interests:
My
research is currently
supported by NSF award DMS
1108754 to study "Steklov
spectra and Divcurl
analysis". The research
centers on the
representation of
solutions of various
linear elliptic problems
with nontrivial boundary
conditions and their
applications to problems
in classical field
theories, gravity and
engineering.
Some
years ago, I developed a
spectral characterization
of trace spaces (see SIMA
Volume 38, (2006),
894907 ) which provides a
constructive approach to
working with boundary
traces of Sobolev
functions. This enables
the use of spectral
methods to describe the
solutions of homogeneous
linear equations Lu = 0 on
a region G subject to
inhomogeneous boundary
conditions Bu = g on the
boundary of G. These
involve finding
representations of
solutions using the
Steklov eigenfunctions of
the operator L on G. This
analysis provides some
different information
about solutions of many
classical linear elliptic
boundary value problems.
It is particularly useful
for understanding
the dependece of solutions
on boundary data.
Since
then I have used these to
study a variety of
phenomena that depend
primarily on the boundary
conditions. A particular
interest is in the
solution of divcurl
boundary value problems on
3dimensional regions.
This is an
overdetermined system of 4
linear equations in 3
unknowns and physically
they are posed subject to
a variety of different
types (and numbers) of
boundary conditions.
The
most important example of
a divcurl system probably
is Maxwell's equation for
electromagnetic fields
but they also arise
in the study of
velocity fields in
fluid mechanics and
are used in computer
graphics for modeling
vector fields.
The
mathematical theory of
divcurl systems is
reasonably well understood
for 2dimensional,
bounded, regions. For
three dimensional
problems, however,
there are still important
open questions since it
is an
overdetermined system of
four linear equations in
three unknowns.
For
different physical
problems divcurl
systems are posed
subject to varying numbers
and types of boundary
conditions;
including mixed b.c.s
where there can be
different numbers of
boundary conditions on
different subsets of the
boundary. The solvability
issues center on
(i)
What are the compatibility
conditions that must hold
for finite energy (L^2)
solutions to exist, and
(ii)
What extra conditions
besides boundary data are
required for the systems
to be wellposed?
The answers to these
questions have interesting
physical and engineering
implications and usually
are based on using an
appropriate variational
principle, and special
choices of "potentials" to
characterize the problem.
Another,
related,
class of problems of
current research interest
is the theory of trace
spaces and problems with
internal interfaces. We
are interested in
finding the
classes of boundary
data that have specific
types of solutions of
certain equations. In
particular this has
led to a description of
trace spaces using Steklov
eigenfunctions. This
spectral theory of such
spaces has advantages in
that it is an intrinisc
theory and there are
explicit formulae for
inner products and
solutions of equations in
terms of these
eigenfunctions.
My work
on these issues is
theoretical mathematics,
involving functional
analysis and variational
principles. I do not
have any funding for
research assistants or
programmers.
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.
