偏微分方程導論

內(nèi)容概要

　　《偏微分方程導論（第2版）》是一部數(shù)學專業(yè)研究生的偏微分方程教程。其旨在讓讀者更好地了解偏微分方程的經(jīng)典基礎結(jié)果，為讀者更深層次學習這方面的專著和教程提供現(xiàn)代理論觀點。這是第二版，較第一版增加了不少練習，專門增加了一章講述擬微分算子，增加了不少材料，內(nèi)容更加豐富。書中的前五章講述經(jīng)典理論，如一階方程，局部存在性定理，數(shù)學物理基礎偏微分方程，適時地運用現(xiàn)代物理技巧解釋長期研究的話題。最后三章專注于現(xiàn)代理論，索伯列夫空間，橢?邊界值問題和擬微分算子。

作者簡介

作者：（美國）福蘭德（Gerald B.Folland）

書籍目錄

PREFACEChapter 0PRELIMINARIESA．Notations and DefinitionsB．Results from Advanced CalculusC．ConvolutionsD．The Fourier TransformE．DistributionsF．Compact OperatorsChapter 1LOCAL EXISTENCE THEoRyA．Basic ConceptsB．Real First Order EquationsC．The General Cauchy ProblemD．The Cauchy-Kowalevski TheoremE．Local Solvability：the Lewy ExampleF．Constant-Coefficient Operators：Fundamental SolutionsChapter 2THE LAPLACE oPERATORA．Symmetry Properties of the LaplacianB．Basic Properties of Harmonic FunotionsC．The Fundamental SolutionD．The Dirichlet and Neumann ProblemsE．The Green’S FunctionF．Dirichlet’S PrincipleG．The Dirichlet Problem in a Half-SpaceH．The Dirichlet Problem in a BallI．More about Harmonic FunctionsChapter 3LAYER PoTENTIALSA．The SetupB．Integral OperatorsC．Double Layer PotentialsD．Single Layer PotentialsE．Solution of the ProblemsF．Further RemarksChapter 4THE HEAT oPERAToRA．The Gaussian KernelB．Funotions of the LaplacianC．The Heat Equation in Bounded DomainsChapter 5THE WAVE oPERAToRA．The Cauchy ProblemB．Solution of the Cauchy ProblemC．The Inhomogeneous EquationD．Fourier Analysis of the Wave OperatorE．The Wave Equation in Bounded DomainsF．The Radon TransformChapter 6THE L2 THEOR．Y oF DERIVATIVESA．Sobolev Spaces on RB．Further Results on Sobolev SpacesC．Local Regularity of Elliptic OperatorsD．Constant-Coefficient Hypoelliptic OperatorsE．Sobolev Spaces on Bounded DomainsChapter 7ELLIPTIC BoUNDARY VALUE PRoBLEMSA．Strong EllipticityB．On Integration by PartsC．Dirichlet Forms and Boundary ConditionsD．The Coercive EstimateE．Existence，Uniqueness，and EigenvaluesF．Regularity at the Boundary：the Second Order CaseG．Further Resuits and TechniquesH．Epilogue：the Return of the Green’S FunctionChapter 8PSEUDODIFFERENTIAL OPERATORSA．Basic Definitions and PropertiesB．Kernels of Pseudodifferential OperatorsC．Asymptotic Expansions of SymbolsD．Amplitudes，Adjoints，and ProductsE．Sobolev EstimatesF．Elliptic OperatorsG．Introduction to Microlocal AnalysisH．Change of CoordinatesBIBLIOGRAPHYINDEX OF SYMBOLSINDEX

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