Harmonic Analysis and PDE

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.