Harmonic Analysis and PDE

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

PI: Sagun Chanillo, Rutgers University DMS-0200628 Abstract: In this proposal several problems are proposed, all in the field of PDE. Most of these problems are in the sub-field of non-linear PDE, and some in linear PDE. The non-linear problems arise naturally from Physics and Geometry and the interaction of Physics and Geometry. The linear problems are also tied in with Geometry and Physics. There is a strong focus in this proposal on various qualitative features of solutions to PDE, their level lines, nodal lines of eigenfunctions, smoothness of solutions and so on. We have selected the problems in part because we view many of them as having very natural connections with classical Harmonic analysis, and in fact we have made some start on solving them using techniques from classical Harmonic analysis. This is described in the body of the proposal. Some of the problems that arise from Physics in our proposal are connected with the phenomena of vortices. Here we continue work that was funded by previous grants. The vortices we study are those that arise from fluid flow on two-dimensional spheres and on the plane. Taking the continuum limit of the point vortex distribution leads us to a new technique for solving the prescribed Gauss curvature equation which also gives a tremendous insight into conformally invariant PDE's. Problems are also posed that stem from the Schrodinger equation and Geometry and its influence on the spectrum of elliptic operators. We single out an important component of the proposed research proposal. In todays environment it has become important to study so-called smart materials, more principally composites. Because of their lightness they are preferred materials to use for their strength and lightness. In this proposal we study as one problem the vibrational characteristics of composites. In particular how should one build objects using composite materials so as to minimize their basic vibrational characteristics. Intuitively the larger the natural frequency with which an object vibrates, the more it is susceptible to stress and breakage. For example one question one can ask is, if we need to build a symmetric object like a washer does it mean the composite has to be arranged symmetrically respecting the symmetry of the washer? We find that in our research for example that this necessarily does not minimize the stresses that can be caused by vibration and we need to arrange the composite in a non-symmetric way to construct the washer. This is not the only problem we study in our proposal but also problems in Differential Geometry and Physics.