Topics in Harnack Inequalities, Degenerate Evolution Equations, and Applied Mathematics

Vanderbilt University, Nashville TN

Investigators

Abstract

Topics in Harnack Inequalities, Degenerate Evolution Equations and Applied Mathematics Abstract of Proposed Research Emmanuele DiBenedetto This project is to continue research on certain important classes of degenerate elliptic and parabolic equations and on homogenization theory. A major topic will be the study of Harnack inequalities. These inequalities have been regarded as properties of the solutions of certain types of linear and non-linear partial differential equations. This project will look at proving the validity of Harnack-type inequalities for new classes of functions. Part of this study will concentrate on the analysis of solutions of parabolic p-Laplacian equations in both the sub-critical and super-critical regimes. The problems in homogenization theory will center on modeling the diffusion of molecules in the rod outer segment (ROS) in the retina of vertebrates. Homogenized limits will be computed and identified for such diffusion processes, set in the cylindrical, thickly layered and incised structures of the ROS. The mathematical problem consists in identifying a topology and relative compactness to permit the limit process. The technical novelty of these structures is that the domain becomes disconnected as the thickness of the layers goes to zero. Harnack inequalities play a central role in the theory of elliptic and parabolic differential equations and are used extensively in analysis, geometry and applications. Determining their range of validity may impact many topics in mathematical analysis. The homogenization problem is motivated by visual transduction in the process of vision. The results of this analysis may cast light on the biological process of vision.