My primary research interests are partial differential equations,nonlinear functional analysis, and numerical analysis. More specifically, my recent investigations have concerned existence, multiplicity, and nodal structure of solutions to Nonlinear Elliptic Boundary Value Problems, using the Variational Method as a primary tool. We are interested in numerical techniques both as a method of approximation and as an experimental tool for investigating the variational structure of related problems. Related interests include: Nonradial Solutions on Radial Regions, Many Sign-Changing Solutions on Specific Regions, More Sign-Changing Solutions on General Regions, Quasilinear Operators such as the p-Laplacian, and Bifurcation Analysis.

In (1), we used variational methods to prove that there exists a solution which changes sign exactly once to the ellipticpartial differential equation \begin{eqnarray*} \left\{ \begin{array}{rl} \Delta u + f(u) = 0 & \hbox{in } \Omega \\ u = 0 & \hbox{in } {\partial \Omega}, \end{array} \right . \end{eqnarray*} where $f$ is subcritical and superlinear; the region $\Omega$ is smooth and bounded in ${\bf R}^n$. Our solution was found to be a Morse Index 2 local minimum of a codimension 2 separating subset of sign-changing functions in the Hilbert space $H_0^{1,2}(\Omega)$. My major contribution and the focus of my Thesis (2) was the construction of a path which realized the min-max at a point in our separating set, which was then shown to be a saddle point and hence a sign-changing solution; the deformation lemma was a key to this proof. This interesting result has been well received, both as a standalone conclusion and as containing ideas which may lead to even more important demonstrations.

Paper (3) contains a numerical algortithm based on the proof of the main theorem in our joint work (1). This article contains several motivating discoveries which have lead either to subsequent results or interesting conjectures. The algorithm includes applications of finite differences, numerical integration, and solving linear systems. The latter is required in order to obtain the Sobolev gradient and currently relies on the SOR method. While the scheme is effective as coded (see the available FORTRAN files at the bottom of this page for demonstrative examples), there is room for improvement in the accuracy and effenciency of the methods used. Following this paper's acceptance, I modified the algorithm to find solutions for a wider class of nonlinearities and on more general regions.

The sign-changing result from (1) appears to have far-reaching consequences. As a result of our variational characterization of sign-changing solutions, my collaborators and I have recently proven in (4) and (6) additional results for asymptotically linear problems and special regions such as balls, disks, and annuli.

Finally, In (5) I followed the method of proof from (1) to obtain a similar result when $f(0)\not=0$, a case which includes so-called Semipositone problems. Of primary interest in this geralization of (1) is that the manifold structure is more complex, lending optimism towards pursuing an even wider class of problems.

My primary future goal is to generalize the structure found in (1) to prove the existence of additional nontrivial solutions of more complex nodal structure resembling higher frequency eigenfunctions, suspected to be higher Morse Index local minima of higher codimension separating subsets. During my PhD research I investigated bifurcation and secondary bifurcation of water waves, radial symmetry, and singular differential equations. For my Masters paper, I solved a system of heat equations numerically and outlined a research plan for solving an inverse problem to determine fluid flow through a porous media. I forsee converging several research directions in the study of parabolic PDEs and their steady state solutions, as well as pursuing some long term projects involving Neural Networks.