Partial Differential Equations and Harmonic Analysis

偏微分方程和调和分析

• 批准号: 9705825
• 负责人: David Jerison
• 金额: --
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Continuing grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-07-01 至 2001-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9705825&HistoricalAwards=false
• 关键词: Partial; Differential; Equations; Harmonic; Analysis

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.

摘要 杰里森 本文的第一个目标是描述三维或多维区域上正曲面上本征函数和绿色函数的水平集。 另一个目标是证明一个独特的连续性，这是适当的替代唯一性的柯西问题的解决方案，椭圆边值问题。 该建议是使用一个最佳的假设上的光滑边界，即Lipschitz正则性。 最优假设使得这成为调和分析中的一个基本问题。 第三个目标是解决两个极值问题的特征值，一个有关的规则性的诺依曼问题，其他旨在证明唯一性的反问题，找到一个域的密度的第一个变化的特征值作为正常变化的函数。 另一个主要目标涉及流体力学激发的三个问题。 第一个问题是要表明，对涡度和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，即经济数据一致性的衡量标准。 第三个是自由边界问题，涉及到用涡线构造平衡流体流动。 最终的目标是研究某些振荡积分与奇异相位，与相机图像中的模式识别的应用。 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，或者导电体的形状如何影响其上电荷的分布。这些问题在数学上密切相关，尽管它们描述了非常不同的物理系统。 温度的情况描述了冰箱或具有绝缘壁的建筑物，但电荷的情况描述了电线和电缆等电气组件。 该项目中由流体力学驱动的部分解决的主要问题是小尺度的拉伸和扭曲如何导致大尺度流体（或空气）的形状，速度和密度的变化。 这是固体力学、塑料力学、弹性材料力学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。 在经济学方面，该项目的最终目标是提供标准，说明何时可以将一组关于购买的经济数据合并成一个有意义的价格指数，尽管存在错误和不一致。 换句话说，不一致性必须有多小才能安全地忽略它们？ 摄像机图像项目将使用傅立叶分析设计一个计算机程序，根据建筑物的一张照片识别建筑物与摄像机视线的角度。 从单个图像进行确定将改进使用两个图像的现有技术，并使计算机视觉更少受到错误的影响。