Harmonic Analysis and Partial Differential Equations

提案：DMS-9971159主要研究者：Andrea R. Nahmod摘要：Nahmod的项目旨在研究调和分析和偏微分方程中的特定问题，这些问题是在域上的光滑条件或操作被放松以符合更自然的物理设置时出现的。第一部分主要研究非规则区域上拉普拉斯算子的边界唯一延拓性质。这个性质是椭圆算子经典的内部唯一延拓性质的一个边界版本，并且与特征函数节点集的结果密切相关。基本上没有什么是已知的非光滑非凸域。本计画将探讨Lipschitz区域是否具有边界唯一连续性。它首先是通过考虑一个特殊的情况，Bang-Bang原则的控制理论，这是感兴趣的本身。第二部分着重于研究双线性算子相关的非光滑符号的背景下出现的补偿紧性，也涉及到卡尔德龙的分解。某些偏微分方程的非线性使得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个困难，利用消除性能和先验界的某些相关的非线性量，通常是双线性。本项目的目的是建立一维非光滑双线性算子映射到某个哈代空间的综合判据。该方法是通过一个微妙的使用时频分析，率先由C。后来被Lacey和Thiele成功利用。该项目的共同目标是开发和获得对分析中涉及的想法和分解的更好理解，目的是使这些技术更容易适用于其他特定环境。偏微分方程是Nahmod项目的最终研究对象，是我们物理世界中许多现象的数学模型。数学分析的作用是研究它们的解的行为，提供提取关于它们的定量和定性信息的工具，并为开发具有合理精度的近似解的方法奠定基础。时频分析是Nahmod采用的关键方法之一，包括将复杂对象分解为局部化且易于理解的基本构建块（通过“调制波形”的集合），然后以简单的方式将它们拼凑在一起。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（播放的瞬间）。完整的“时频合成”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出现的更抽象的振荡表达式。这种类型的分析与小波理论密切相关。它的影响是理论和计算的潜在实施的想法，在谐波分析产生快速计算算法的操作，由于其非本地的性质，否则昂贵的数值计算。

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.