Partial Differential Equations and Harmonic Analysis

研究的第一个目标是描述三维或多维区域和正曲面上的本征函数和格林函数的水平集。另一个目的是证明一个唯一的连续性，它是椭圆边值问题解的柯西问题唯一性的适当替代。建议使用关于边界光滑性的最优假设，即Lipschitz正则性。最优假设使这成为调和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Neumann问题中的正则性有关，另一个目的是证明在给定特征值的第一个变量的密度作为正态变量的函数的情况下，寻找区域的反问题的唯一性。另一个主要目标与流体力学引发的三个问题有关。第一个问题是证明涡度和可压缩性的无穷小控制蕴含着某种全局控制。第二个问题与第一个问题有关，即衡量经济数据的一致性。第三个问题是与构造涡线平衡流体流动有关的自由边界问题。最后的目标是研究某些具有奇异位相的振荡积分，并将其应用于摄像机图像的模式识别。主要的项目是了解一个区域的形状如何影响该区域内的热量分布，或者电导体的形状如何影响其上的电荷分布。这些问题在数学上密切相关，尽管它们描述的是非常不同的物理系统。温度描述的是冰箱或有隔热墙的建筑物，而电荷描述的是电线和电缆等电子部件。由流体力学推动的项目部分讨论的主要问题是，在小范围内拉伸和扭曲如何导致大范围内流体(或空气)的形状、速度和密度发生变化。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了类似的数学模型。在经济学的背景下，该项目的最终目标是给出关于购买的一组经济数据何时可以被组合成有意义的价格指数的标准，尽管存在错误和不一致。换句话说，不一致必须有多小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，根据建筑物的一张照片来识别建筑物与相机视线的夹角。根据一张图像进行确定将改进使用两张图像的现有技术，并使计算机视觉更少受到错误的影响。

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.