### 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 types (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.

Current Address: Department of Mathematics, PGH Building, University of Houston, Houston, Texas 77204-3008
Phone(UH): (713) 743-3500 - Fax(UH): (713) 743-3505