Partial Differential Equations and Harmonic Analysis

本文提出的第一个研究目标是描述三维或多维区域和正曲面上的特征函数和格林函数的水平集。另一个目的是证明椭圆型边值问题解的唯一性，该唯一性可以代替柯西问题解的唯一性。建议采用边界光滑性的最优假设，即Lipschitz规则。最优假设使其成为谐波分析中的一个基本问题。第三个目标是解决特征值的两个极值问题，一个与诺伊曼问题中的正则性有关，另一个旨在证明在给定特征值的第一个变化的密度作为正态变化的函数的情况下寻找一个逆问题中的唯一性。另一个主要目标涉及由流体力学引起的三个问题。第一个问题是证明对涡量和可压缩性的无穷小控制意味着某种全局控制。第二个问题与第一个问题有关，即衡量经济数据一致性的标准。第三是关于构造带涡线的平衡流体流动的自由边界问题。最终目标是研究具有奇异相位的振荡积分，并将其应用于相机图像的模式识别。主要项目是了解一个区域的形状如何影响该区域的热量分布，或者电导体的形状如何影响其上电荷的分布。这些问题在数学上是密切相关的，尽管它们描述的是非常不同的物理系统。温度的情况是指冰箱或有绝缘墙壁的建筑物，而电荷的情况是指电线和电缆等电子元件。在流体力学驱动的项目部分中，主要解决的问题是如何在小尺度上拉伸和扭曲导致大尺度上流体（或空气）的形状、速度和密度的变化。这是固体、塑料、弹性材料以及流体力学中的一个基本问题。在经济均衡理论中也遇到类似的数学模型。在经济学背景下，这个项目的最终目标是给出一个标准，当一组关于购买的经济数据可以组合成一个有意义的价格指数时，尽管存在错误和不一致。换句话说，矛盾必须小到什么程度，人们才能安全地忽略它们？相机图像项目将使用傅里叶分析来设计一个计算机程序，以识别基于建筑物的单张照片的相机视线与建筑物的角度。单幅图像的确定将改进现有的使用两幅图像的技术，使计算机视觉更少受到误差的影响。

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.