Harmonic Analysis and Partial Differential Equations

调和分析和偏微分方程

• 批准号: 9971159
• 负责人: Andrea Nahmod
• 金额: $ 7.08万
• 依托单位: University of Massachusetts Amherst
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-07-01 至 2002-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971159&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Partial; Differential; Equations

Proposal: DMS-9971159Principal Investigator: Andrea R. NahmodAbstract: Nahmod's project aims at studying specific problems in harmonic analysis and partial differential equations that arise when smoothness conditions on the domain or the operations are relaxed to conform with a more natural physical setting. The first part concentrates on the study of the boundary unique continuation property for the Laplace operator on irregular domains. This property is a boundary version of the classical interior unique continuation property for elliptic operators, and is closely related to results on nodal sets of eigenfunctions. Basically nothing is known for nonsmooth-nonconvex-domains. This project will investigate whether or not Lipschitz domains have the boundary unique continuation property. It is first approached by considering a special case, the Bang-Bang Principle of control theory, which is of interest in its own right. The second part focuses on the study of bilinear operators associated to nonsmooth symbols that arise in the context of compensated compactness and are also related to Calderon's commutators. Nonlinearity in certain partial differential equations prevents direct use of weak continuity arguments to ensure the convergence of approximating solutions. Compensated compactness was developed to overcome this difficulty by exploiting cancellation properties and a priori bounds of certain associated nonlinear quantities, usually bilinear. The aim of this project is to establish a comprehensive criterion in one dimension for bilinear operators with nonsmooth symbols to map into some Hardy space. The approach is via a delicate use of time-frequency analysis, as pioneered by C. Fefferman and later successfully exploited by Lacey and Thiele. A concurrent aim of the project is to both develop and gain a greater understanding of the ideas and decompositions involved in the analysis with the goal of making these techniques more readily applicable in other specific contexts. Partial differential equations, the ultimate object of study in Nahmod's project, are the mathematical models of the laws governing many phenomena in our physical world. The role of mathematical analysis is to study the behaviour of their solutions, provide the tools to extract quantitative and qualitative information about them, and lay the foundations upon which methods to approximate the solutions with reasonable accuracy are developed. Time-frequency analysis, one of the key methods to be employed by Nahmod, consists in decomposing complex objects into basic building blocks (via a collection of "modulated waveforms") that are localized and easy to understand, and then piecing them back together in a straightforward manner. A time-frequency analysis of a problem can be likened to a musical score. The modulated waveforms (the notes) have four attributes: amplitude (loudness), scale (duration), frequency (pitch) and position (instant it is played). The object that the full "time-frequency composition" portrays might, for example, be speech, a radar signal, or a fingerprint, but might also be a more abstract oscillatory expression arising in optics, acoustic scattering, and wave propagation problems. This type of analysis is closely related to the theory of wavelets. Its impact is both theoretical and computational for its potential to implement the ideas developed in harmonic analysis to produce fast computational algorithms for operations which, due to their nonlocal nature, are otherwise expensive to compute numerically.

提案：DMS-9971159原理研究者：Andrea R. Nahmodabstract：Nahmod的项目旨在研究谐波分析和偏微分方程中的特定问题，当域上的平滑性条件或操作放松时，这些问题会出现，以使其与更自然的物理环境相吻合。第一部分集中于对不规则域上拉普拉斯操作员的边界独特延续性质的研究。该属性是椭圆运算符的经典内部独特延续属性的边界版本，并且与特征函数的节点集的结果密切相关。基本上，没有什么以非滑孔convex-domains闻名的。该项目将调查Lipschitz域是否具有边界独特的延续属性。首先，考虑了一种特殊情况，即控制理论的爆炸原则，这本身就是感兴趣的。第二部分的重点是研究与不平滑符号相关的双线性操作员，这些符号是在补偿紧凑型的背景下出现的，也与Calderon的换向器有关。在某些部分偏微分方程中的非线性可阻止直接使用弱连续性参数，以确保近似解决方案的收敛性。开发了补偿的紧凑性来克服这一困难，通过利用取消特性和某些相关的非线性量（通常是双线性）的先验界限。该项目的目的是在一个维度上建立一个综合标准，以使双线性操作员具有非平滑符号，以将其映射到一些耐寒的空间中。该方法是通过C. fefferman的率先使用时频分析的精致使用，后来莱西和Thiele成功地利用了这种分析。该项目的同时目的是既要对分析中涉及的思想和分解有更深入的了解，以使这些技术更容易适用于其他特定情况。部分微分方程是Nahmod项目中研究的最终对象，是管理我们物理世界中许多现象的法律的数学模型。数学分析的作用是研究其解决方案的行为，提供有关提取定量和定性信息的工具，并奠定基础，以合理的精度近似于解决方案的方法。频率分析是Nahmod要采用的关键方法之一，是将复杂的对象分解为基本的构建块（通过“调制波形”的集合），这些对象本地化且易于理解，然后以简单的方式将它们拼凑在一起。对问题的时频分析可以比作音乐得分。调制的波形（注释）具有四个属性：振幅（响度），比例（持续时间），频率（音高）和位置（即时播放）。完整的“时频组成”所描绘的对象可能是语音，雷达信号或指纹，但也可能是在光学，声学散射和波浪传播问题中引起的更抽象的振荡表达。这种类型的分析与小波理论密切相关。它的影响既是理论上的，又是计算的潜力，即实施在谐波分析中开发的思想，以生成快速计算算法的操作算法，由于其非本地性质，以数值来计算，这些算法却是昂贵的。