Harmonic Analysis and Partial Differential Equations

摘要：Nahmod的项目旨在研究谐波分析和偏微分方程中的特定问题，当域上的平滑条件或操作被放宽以符合更自然的物理环境时。第一部分主要研究了不规则域上拉普拉斯算子的边界惟一延拓性质。该性质是椭圆算子经典内唯一延拓性质的边界版本，与本征函数节点集上的结果密切相关。基本上，对于非光滑-非凸域，我们一无所知。本课题将探讨Lipschitz域是否具有边界唯一延拓性。首先考虑一个特殊情况，即控制理论的Bang-Bang原理，它本身就很有趣。第二部分重点研究补偿紧性背景下与非光滑符号相关的双线性算子，这些算子也与Calderon换向子有关。由于某些偏微分方程的非线性，不能直接使用弱连续性参数来保证近似解的收敛性。补偿紧性是为了克服这一困难而发展起来的，它利用了某些相关非线性量（通常是双线性的）的消去性质和先验界。本课题的目的是在一维上建立具有非光滑符号的双线性算子映射到Hardy空间的一个综合准则。这种方法是通过对时频分析的巧妙运用，这种方法由C. Fefferman首创，后来被Lacey和Thiele成功地利用。该项目的另一个共同目标是开发并获得对分析中涉及的思想和分解的更深入的理解，目标是使这些技术更容易适用于其他特定的上下文中。偏微分方程是Nahmod项目的最终研究对象，它是支配我们物理世界中许多现象的规律的数学模型。数学分析的作用是研究它们的解的行为，提供工具来提取关于它们的定量和定性信息，并奠定基础，在此基础上开发出具有合理精度的近似解的方法。时频分析是Nahmod使用的关键方法之一，它将复杂的对象分解成基本的构建块（通过一组“调制波形”），这些构建块是局部的，易于理解的，然后以一种直接的方式将它们拼凑在一起。对问题的时频分析可以比作乐谱。调制波形（音符）有四个属性：振幅（响度）、音阶（持续时间）、频率（音高）和位置（演奏的瞬间）。例如，完整的“时频组合”描绘的对象可能是语音、雷达信号或指纹，但也可能是光学、声散射和波传播问题中产生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。它的影响是理论的和计算的，因为它有可能实现谐波分析中发展的思想，为操作产生快速的计算算法，由于它们的非局部性质，否则在数值计算上是昂贵的。

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.