Research interests:
My research is currently supported by NSF award DMS
0808115 to study "Div-curl systems and Variational Principles". The
research uses variational principles, and methods, to prove results
about fundamental equations from continuum mechanics and classical
field theories.
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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