Harmonic Analysis and PDE

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PI: Sagun Chanillo, Rutgers UniversityDMS-0200628Abstract:In this proposal several problems are proposed, all in the field ofPDE. Most of these problems are in the sub-field of non-linearPDE, and some in linear PDE. The non-linear problemsarise naturally from Physics andGeometry and the interaction of Physics and Geometry. The linear problems are also tied in withGeometry and Physics. Thereis a strong focus in this proposal on various qualitativefeatures of solutions to PDE, their level lines, nodal linesof 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 ofvortices. Here we continue work that was funded by previous grants.The vortices we study are those that arise from fluid flow ontwo-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 posedthat stem from the Schrodinger equation and Geometry and itsinfluence on the spectrum of elliptic operators.We single out an important component of the proposed researchproposal. In todays environment it has become important tostudy so-called smart materials, more principally composites.Because of their lightness they are preferred materialsto use for their strength and lightness. In this proposalwe study as one problem the vibrational characteristicsof composites. In particular how should one build objects usingcomposite materials so as to minimize their basic vibrational characteristics.Intuitively the larger the natural frequency with whichan object vibrates, the more it is susceptible to stressand breakage. For example one question one can ask is, if we need to build a symmetric object like a washer does itmean the composite has to be arranged symmetrically respectingthe symmetry of the washer? We find that in our research for example thatthis necessarily does not minimize the stresses that can be caused by vibrationand we need to arrange the composite in a non-symmetric wayto construct the washer. This is not the only problem we study in ourproposal but also problems in Differential Geometry and Physics.