Research interests:
My research is currently supported by NSF award DMS
1108754 to study "Steklov spectra and Div-curl analysis". The
research centers on the representation of solutions of various linear
elliptic problems with non-trivial 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), 894-907 ) 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 div-curl boundary value problems on 3-dimensional
regions.
This is an overdetermined system of 4 linear equations in 3
unknowns and physically they are posed subject to a variety of
different tyes (and numbers) of boundary conditions.
The most important example of a div-curl
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 div-curl systems is reasonably well understood
for
2-dimensional, bounded, regions. For three dimensional problems,
however,
there are still important open questions since it is an
over-determined
system of four linear equations in three unknowns.
For different
physical problems div-curl 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 well-posed?
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.
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