線性偏微分算子分析 第1卷 第2版(英文影印版)

內(nèi)容概要

　　The progress in the theory of linear partial differential equations during the past 30 years owes much to the theory of distributionscreated by Laurent Schwartz at the end of the 1940's. It summed up agreat deal of the experience accumulated in the study of partial differ-ential equations up to that time， and it has provided an ideal frame-work for later developments. "Linear partial differential operators" be-gan with a brief summary of distribution theory for this was still un-familiar to many analysts 20 years ago. The presentation then pro-ceeded directly to the most general results available on partial differ-ential operators. Thus the reader was expected to have some prior fa-miliarity with the classical theory although it was not appealed to ex- plicitly. Today it may no longer be necessary to include basic distribu-tion theory but it does not seem reasonable to assume a classical background in the theory of partial differential equations since mod- ern treatments are rare. Now the techniques developed in the study of singularities of solutions of differential equations make it possible to regard a fair amount of this material as consequences of extensions of distri'oution theory. Rather than omitting distribution theory I have therefore decided to make the first volume of this book a greatly ex-　panded treatment of it. The title has been modified so that it indicates the general analytical contents of this volume.　　……

書籍目錄

Introduction ChapterⅠ Test Functions   Summary   1.1 A review of Differential Calculus   1.2 Existence of Test Functions   1.3 Convolution   1.4 Cutoff Functions and Partitions of Unity   Notes ChapterⅡ Definition and Basic Properties of Distributions   Summary   2.1 Basic Definitions   2.2 Localization   2.3 Distributions with Compact Support   Notes ChapterⅢ Differentiation and Multiplication by Functions   Summary   3.1 Definition and Examples   3.2 Homogeneous Distributions   3.3 Some Fundamental Solutions   3.4 Evaluation of Some Integrals   NotesChapterⅣ  Convolution  Summary  4.1 Convolution with a Smooth Function  4.2 Convolution of Distributions  4.3 The Theorem of Supports  4.4 The Role of Fundamental Solutions  4.5 Basic Lp Estimates for Convolutions  NotesChapterⅤ  Distributions in Product Spaces  Summary  5.1 Tensor Products  5.2 The Kernel Theorem  NotesChapterⅥ  Composition with Smooth Maps  Summary  6.1 Definitions  6.2 Some Fundamental Solutions  6.3 Distributions on a Manifold  6.4 The Tangent and Cotangent Bundles  NotesChapterⅦ  The Fourier Transformation  Summary  7.1 The Fourier Transformation in and in   7.2 Poissons Summation Formula and Periodic Distributions  7.3 The Fourier-Laplace Transformation in   7.4 More General Fourier-Laplace Transforms  7.5 The Malgrange Preparatio Theorem  7.6 Fourier Transforms of Gaussian Functions  7.7 The Method of Stationary Phase  7.8 Oscillatory Integrals  7.9 H(s)Lp and Holder Estimates  NotesChapterⅧ  Spectral Analysis of Singularities  Summary  8.1 The Wave Front Set  8.2 A Review of Operations with Distributions  8.3 The Wave Front Set of Solutions Of Partial Differential Equations  8.4 THe Wave Front Set With Respect to   8.5 Rules of Computation for WFL  8.6 WFL for Solutions of Partial Differential Equations  8.7 Microhyperbolicity  NotesChapterⅨ  Hyperfunctions  Summary  ……ExercisesAnswers and Hints to All the ExercisesBibliographyIndexIndex of Notation

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