Series Preface Preface 1 Introduction 1.1 Basic Mathematical Questions 1.1.1 Existence 1.1.2 Multiplicity 1.1.3 Stability 1.1.4 Linear Systems of ODEs and Asymptotic Stability 1.1.5 Well-Posed Problems 1.1.6 Representations 1.1.7 Estimation 1.1.8 Smoothness 1.2 Elementary Partial Differential Equations 1.2.1 Laplace's Equation 1.2.2 The Heat Equation 1.2.3 The Wave Equation 2 Characteristics 2.1 Classification and Characteristics 2.1.1 The Symbol of a Differential Expression 2.1.2 Scalar Equations of Second Order 2.1.3 Higher-Order Equations and Systems 2.1.4 Nonlinear Equations 2.2 The Cauchy-Kovalevskaya Theorem 2.2.1 Real Analytic Punctions 2.2.2 Majorization 2.2.3 Statement and Proof of the Theorem 2.2.4 Reduction of General Systems 2.2.5 A PDE without Solutions 2.3 Holmgren's Uniqueness Theorem 2.3.1 An Outline of the Main Idea 2.3.2 Statement and Proof of the Theorem 2.3.3 The Weierstrafi Approximation Theorem 3 Conservation Laws and Shocks 3.1 Systems in One Space Dimension 3.2 Basic Definitions and Hypotheses 3.3 Blowup of Smooth Solutions 3.3.1 Single Conservation Laws 3.3.2 The p System 3.4 Weak Solutions 3.4.1 The Rankine-Hugoniot Condition 3.4.2 Multiplicity 3.4.3 The Lax Shock Condition 3.5 Riemann Problems 3.5.1 Single Equations 3.5.2 Systems 3.6 Other Selection Criteria 3.6.1 The Entropy Condition 3.6.2 Viscosity Solutions 3.6.3 Uniqueness 4 Maximum Principles 4.1 Maximum Principles of Elliptic Problems 4.1.1 The Weak Maximum Principle 4.1.2 The Strong Maximum Principle 4.1.3 A Priori Bounds 4.2 An Existence Proof for the Dirichlet Problem 4.2.1 The Dirichlet Problem on a Ball 4.2.2 Subharmonic Functions 4.2.3 The Arzela-Ascoli Theorein 4.2.4 Proof of Theorem 4.13 4.3 Radial Symmetry 4.3.1 Two Auxiliary Lemmas 4.3.2 Proof of the Theorem 4.4 Maximum Principles for Parabolic Equations 4.4.1 The Weak Maximum Principle 4.4.2 The Strong Maximum Principle 5 Distributions 5.1 Test Functions and Distributions 5.1.1 Motivation 5.1.2 Test Functions 5.1.3 Distributions 5.1.4 Localization and Regularization 5.1.5 Convergence of Distributions 5.1.6 Tempered Distributions 5.2 Derivatives and Integrals 5.2.1 Basic Definitions 5.2.2 Examples 5.2.3 Primitives and Ordinary Differential Equations 5.3 Convolutions and Fundamental Solutions 5.3.1 The Direct Product of Distributions 5.3.2 Convolution of Distributions 5.3.3 Fundamental Solutions 5.4 The Fourier Transform 5.4.1 Fourier Transforms of Test Functions 5.4.2 Fourier Transforms of Tempered Distributions 5.4.3 The Fundamental Solution for the Wave Equation 5.4.4 Fourier Transform of Convolutions 5.4.5 Laplace Transforms 5.5 Green's Functions 5.5.1 Boundary-Value Problems and their Adjoints 5.5.2 Green's Functions for Boundary-Value Problems 5.5.3 Boundary Integral Methods 6 Function Spaces 6.1 Banach Spaces and Hilbert Spaces 6.1.1 Banach Spaces 6.1.2 Examples of Banach Spaces 6.1.3 Hilbert Spaces 6.2 Bases in Hilbert Spaces 6.2.1 The Existence of a Basis 6.2.2 Fourier Series 6.2.3 Orthogonal Polynomials 6.3 Duality and Weak Convergence 6.3.1 Bounded Linear Mappings 6.3.2 Examples of Dual Spaces 6.3.3 The Hahn-Banach Theorem 6.3.4 The Uniform Boundedness Theorem 6.3.5 Weak Convergence 7 Sobolev Spaces 7.1 Basic Definitions 7.2 Characterizations of Sobolev Spaces 7.2.1 Some Comments on the Domain Ω 7.2.2 Sobolev Spaces and Fourier Transform 7.2.3 The Sobolev Imbedding Theorem 7.2.4 Compactness Properties 7.2.5 The Trace Theorem 7.3 Negative Sobolev Spaces and Duality 7.4 Technical Results 7.4.1 Density Theorems 7.4.2 Coordinate Transformations and Sobolev Spaces on Manifolds 7.4.3 Extension Theorems 7.4.4 Problems 8 Operator Theory 8.1 Basic Definitions and Examples 8.1.1 Operators 8.1.2 Inverse Operators 8.1.3 Bounded Operators,Extensions 8.1.4 Examples of Operators 8.1.5 Closed Operators 8.2 The Open Mapping Theorem 8.3 Spectrum and Resolvent 8.3.1 The Spectra of Bounded Operators 8.4 Symmetry and Self-adjointness 8.4.1 The Adjoint Operator 8.4.2 The Hilbert Adjoint Operator 8.4.3 Adjoint Operators and Spectral Theory 8.4.4 Proof of the Bounded Inverse Theorem for Hilbert Spaces 8.5 Compact Operators 8.5.1 The Spectrum of a Compact Operator 8.6 Sturm-Liouvillc Boundary-Value Problems 8.7 The Fredholm Index 9 Linear Elliptic Equations 9.1 Definitions 9.2 Existence and Uniqueness of Solutions of the Dirichlet Problem 9.2.1 The Dirichlet Problem-Types of Solutions 9.2.2 The Lax-Milgram Lemma 9.2.3 G*rding's Inequality 9.2.4 Existence of Weak Solutions 9.3 Eigenfunction Expansions 9.3.1 Fredholm Theory 9.3.2 Eigenfunction Expansions 9.4 General Liuear Elliptic Problems 9.4.1 The Neumann Problem 9.4.2 The Complementing Condition for Elliptic Systems 9.4.3 The Adjoint Boundary-Value Problem 9.4.4 Agmon's Condition and Coercive Problems 9.5 Interior Regularity 9.5.1 Difference Quotients 9.5.2 Second-Order Scalar Equations 9.6 Boundary Regularity 10 Nonlinear Elliptic Equations 10.1 Perturbation Results 10.1.1 The Banach Contraction Principle and the Implicit Function Theorem 10.1.2 Applications to Elliptic PDEs 10.2 Nonlinear Variational Problems 10.2.1 Convex problems 10.2.2 Nonconvex Problems 10.3 Nonlinear Operator Theory Methods 10.3.1 Mappings on Finite-Dimensional Spaces 10.3.2 Monotone Mappings on Banach Spaces 10.3.3 Applications of Monotone Operators to Nonlinear PDEs 10.3.4 Nemytskii Operators 10.3.5 Pseudo-monotone Operators 10.3.6 Application to PDEs 11 Energy Methods for Evolution Problems 11.1 Parabolic Equations 11.1.1 Banach Space Valued Functions and Distributions 11.1.2 Abstract Parabolic Initial-Value Problems 11.1.3 Applications 11.1.4 Regularity of Solutions 11.2 Hyperbolic Evolution Problems 11.2.1 Abstract Second-Order Evolution Problems 11.2.2 Existence of a Solution 11.2.3 Uniqueness of the Solution 11.2.4 Continuity of the Solution 12 Semigroup Methods 12.1 Semigroups and Infinitesimal Generators 12.1.1 Strongly Continuous Semigroups 12.1.2 The Infinitesimal Generator 12.1.3 Abstract ODEs 12.2 The Hille-Yosida Theorem 12.2.1 The Hille-Yosida Theorem 12.2.2 The Lumer-Phillips Theorem 12.3 Applications to PDEs 12.3.1 Symmetric Hyperbolic Systems 12.3.2 The Wave Equation 12.3.3 The Schrödinger Equation 12.4 Analytic Semigroups 12.4.1 Analytic Semigroups and Their Generators 12.4.2 Fractional Powers 12.4.3 Perturbations of Analytic Semigroups 12.4.4 Regularity of Mild Solutions A References A.1 Elementary Texts A.2 Basic Graduate Texts A.3 Specialized or Advanced Texts A.4 Multivolume or Encyclopedic Works A.5 Other References Index
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