Tools for Pde: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
Tools for Pde: Pseudodifferential Operators, Paradifferential Operators, and Layer Potentials
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DOI：
10.1090/surv/081
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发表时间：
2000-07
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通讯作者：
Michael Taylor
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作者：
Michael Taylor
这本书开发了三个相关的工具，在偏微分方程的分析是有用的，从奇异积分算子的经典研究产生：伪微分算子，准微分算子，和层势。贯穿整个工作的主题是在相对较少规律性的情况下治疗PDE。第一章研究了一类符号具有有限正则度的伪微分算子；第二章展示了准微分算子如何对各种非线性算子在函数空间上的作用给出尖锐的估计。第三章将这些材料应用于PDE的各种结果，包括粗糙系数椭圆PDE的正则性结果，粗糙区域上的平面流体流动，里奇张量上给定弱界的黎曼流形的估计，div-旋度估计，以及粗糙系数波动方程奇点传播的结果。最后一章研究了Lipschitz域上的层势方法，重点讨论了变系数椭圆偏微分方程边界问题的应用。Michael Taylor是几本关于偏微分方程和伪微分算子主题的知名书籍的作者。他的非交换谐波分析，第22卷在数学调查和专著系列出版的AMS，是一个很好的介绍使用李群在线性分析和偏微分方程。目前的书，工具的PDE，是适合作为一个文本的高级研究生准备集中在PDE和/或谐波分析。
This book develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential operators, paradifferential operators, and layer potentials. A theme running throughout the work is the treatment of PDE in the presence of relatively little regularity. The first chapter studies classes of pseudodifferential operators whose symbols have a limited degree of regularity; the second chapter shows how paradifferential operators yield sharp estimates on the action of various nonlinear operators on function spaces. The third chapter applies this material to an assortment of results in PDE, including regularity results for elliptic PDE with rough coefficients, planar fluid flows on rough domains, estimates on Riemannian manifolds given weak bounds on Ricci tensor, div-curl estimates, and results on propagation of singularities for wave equations with rough coefficients. The last chapter studies the method of layer potentials on Lipschitz domains, concentrating on applications to boundary problems for elliptic PDE with variable coefficients. Michael Taylor is the author of several well-known books on topics in PDEs and pseudodifferential operators. His Noncommutative Harmonic Analysis, Volume 22 in the Mathematical Surveys and Monographs series published by the AMS, is a good introduction to the use of Lie groups in linear analysis and PDEs. The present book, Tools for PDE, is suitable as a text for advanced graduate students preparing to concentrate in PDE and/or harmonic analysis.