偏微分方程IV
2009-1
科学出版社
叶戈罗夫
241
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要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。 从出版方面来讲，除了较好较快地出版我们自己的成果外，引进国外的先进出版物无疑也是十分重要与必不可少的。从数学来说，施普林格（springer）出版社至今仍然是世界上最具权威的出版社。科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。 这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。 当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。 总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。
This volume of the Encyclopaedia contains two contributions．In the first Yu.V.Egorov gives an accomnt of microlocal analysis as a tool for investigating partial differemial equations．This 113ethod has become increasingly important in the theory．of Hamiltonian systems in recent years． The second survey written by V.Ya．1vrii treats linear hyperbolic equations and systems．The author states necessary and sufficiient conditions for C∞-and L2-well-posedness and he studies the analogous pmhlem in the comext ofGevrey classes．He also describes,the latest results in，the theory of mixed problems for hyperbolic operators and concludes with a list of unsolved problems． Both parts coyer recent research in two important fields,which before was scattered in numerous joumals．The book will hence be of immense value to graduate students and researchers in partial differential equationS and theoretical physics。
Chapter 1.Microlocal Properties of Distributions 2.Wave Front of Distribution.Its Functorial Properties 2.1.Definition ofthe Wave Front 2.2.Localization ofWave Front 2.3.Wave Front and Singularities of One—Dimensional 2.4.Wave Fronts of Pushforwards and Pullbacks of a 3.Wave Front and Operations on Distributions 3.1 The Trace of a Distribution.Product of Distnritbiaul Eiuation 3.2.The Wave Front of the Solution of a Differential Eqution 3.3.Wave Fronts and Integral OperatorsChapter 2.Pseudodifferential Operators 1.Algebra ofPseudodifferential Operators 1.1.Singular Integral Operators 1.2.The Symbol 1.3.Boundedness of Pseudodifferential Operators 1.4.Composition of Pseudodifferential Operators 1.5.The Formally Adjoint Operator 1.6.Pseudolocality.Microlocality 1.7.Elliptic Operators 1.8.Garding’S Inequality 1.9.Extension 0f the Class of Pseudodifferential Operators 2.Invariance of the Principal SymboJ Under Canonical Transformations 2.1.Invariance Under the Change ofVariables. 2.2 The Subprincipal Symbol 2.3.Canonical Transformations 2.4.An Inverse Theorem 3.Canonical Forms ofthe Symbol 3.1.Simple Characteristic Points 3.2.Double Characteristics 3.3.The Complex-alued Symbol 3.4.The Canonical Form of the Symbol in a Neighbourhood of the Boundary. 4.Various Classes of Pseudodifferential Operators 4.1.The Lm/pδClasses 4.2.The Lm/φ,φ Classes 4.3。The Weyl Operators 5.Complex Powers ofElliptic Operators 5.1.The Definition ofComplex Powers. 5.2.Thc Construction of the Symbol for the Operator Az 5.3.The Construction of the Kernel of the Operator Az 5.4.The ξ-Function ofan Elliptic Operator 5.5.The Asymptotics of the Spectral Function and Eigenvalues 5.6.Complex Powers of an Elliptic Operator with Boundary Conditions 6.Pseudodifferential Operators in IRn and Quantization 6.1.The Analogy Between the Microlocal Analysis and the Quantization 6.2.Pseudodifierential 0perators in RnChapter 3.Fourier Integral Operators 1.The Parametrix of the Cauchy Problem for Hyperbolic Equations 1.1.The Cauchy Problem for the Wave Equation 1.2.The Cauchy Problem for the Hyperbolic Equation of an Arbitrary 0rder. 1.3.The Method of Stationary Phase 2.The Maslov Canonical Operator 2.1.The MaslOV Index 2.2.Pre.canonieal Operator 2.3.The Canonical Operator 2.4.Some Applications. 3.Fourier Integral Operators 3.1.The Oscillatory Integrals 3.2.The Local Definition of the Fourier Integral Operator ……Chapter 4 The Propagation of SingularitiesChapter 5 Solvbility of (Pseudo)Differential EquationsChapter 6 Smoothness of Solutions of Differential EquationsChapter 7 Transformation of Boundary-Value ProblemsChapter 8 HyperfuctionsReferences
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