Partial Differential Equations and Harmonic Analysis

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Abstract Jerison The first goal of the research proposed is to describe level sets of eigenfunctions and Green's function on regions in three or more dimensions and on positively curved surfaces. Another goal is to prove a unique continuation property which is the appropriate substitute for uniqueness in the Cauchy problem for solutions to elliptic boundary value problems. The proposal is to use an optimal hypothesis on the smoothness of the boundary, namely, Lipschitz regularity. The optimal hypothesis makes this a fundamental issue in harmonic analysis. A third goal is to solve two extremal problems for eigenvalues, one related to regularity in the Neumann problem, the other aimed at proving uniqueness in an inverse problem of finding a domain given the density of the first variation of the eigenvalue as a function of normal variation. Another main goal concerns three problems motivated by fluid mechanics. The first problem is to show that infinitesimal control on vorticity and compressibility implies some global control. The second problem relates the first to measures of consistency of economic data. The third is a free boundary problem related to constructing equilibrium fluid flows with vortex lines. The final goal is to study certain oscillatory integrals with singular phases, with an application to pattern recognition in camera images. The main project is to understand how the shape of a region affects the distribution of heat in the region or how the shape of an electrical conductor affects the distribution of electrical charge on it. These problems are closely related mathematically, even though they describe very different physical systems. The case of temperature describes a refrigerator or building with insulated walls, but the case of electrical charge describes electrical components like wires and cables. The main issue addressed in the part of the project motivated by fluid mechanics is how stretching and twisting at small scales leads to changes in the shape, speed, and density of a fluid (or air) at large scales. This is a fundamental question in the mechanics of solids, plastics, and elastic materials, as well as fluids. A similar mathematical model is encountered in economic equilibrium theories. The ultimate goal of the project in the economics context is to give criteria for when a set of economic data about purchases can be combined into a meaningful price index, despite errors and inconsistency. In other words, how small must inconsistencies be in order that one can safely ignore them? The camera image project will use Fourier analysis to design a computer program to recognize the angle a building makes with the camera's line of sight based on a single photograph of the building. The determination from a single image would improve on existing techniques using two images and make computer vision less subject to error.