Harmonic Analysis and Partial Differential Equations

建议：DMS-9971159首席研究员：Andrea R.Nahmod的项目旨在研究调和分析和偏微分方程中的特定问题，这些问题是当域或操作上的光滑性条件放宽以符合更自然的物理环境时出现的。第一部分主要研究了非规则区域上Laplace算子的边界唯一延拓性。这一性质是椭圆算子的经典内部唯一连续性质的边界版本，并且与特征函数节点集上的结果密切相关。对于非光滑非凸域，基本上没有什么是已知的。本课题将研究Lipschitz域是否具有边界唯一连续性质。它首先是通过考虑一种特殊情况来探讨的，即控制理论的Bang-Bang原理，它本身就是有意义的。第二部分主要研究了在补偿紧性的背景下产生的与非光滑符号相关的双线性算子以及与Calderon交换子有关的双线性算子。某些偏微分方程解的非线性使得不能直接使用弱连续变量来保证逼近解的收敛。补偿紧性的提出是为了克服这一困难，它利用抵消性质和某些相关的非线性变量的先验界，通常是双线性的。本课题的目的是建立一维空间中具有非光滑符号的双线性算子映射到某个Hardy空间的综合准则。这种方法是通过巧妙地使用时频分析来实现的，正如C.Fefferman所开创的，后来被Lacey和Thiele成功利用。该项目的同时目标是发展和更好地理解分析中涉及的想法和分解，目的是使这些技术更容易适用于其他具体情况。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定律的数学模型。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，并为制定以合理精度逼近解的方法奠定基础。时频分析是Nahmod要使用的关键方法之一，它包括将复杂对象分解成局部且易于理解的基本构建块(通过一组“调制波形”)，然后以直接的方式将它们重新拼接起来。问题的时频分析可以比作乐谱。调制的波形(音符)具有四个属性：幅度(响度)、音阶(持续时间)、频率(音调)和位置(即时播放)。完整的“时频成分”所描绘的对象可能是，例如，语音、雷达信号或指纹，但也可能是在光学、声散射和波传播问题中产生的更抽象的振荡表达。这类分析与小波理论密切相关。它的影响既是理论上的，也是计算上的，因为它有可能实现调和分析中发展的思想，为运算产生快速计算算法，因为它们的非局部性质，否则数值计算成本很高。

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.