Pub. Date:
Springer Berlin Heidelberg
Foundations of the Classical Theory of Partial Differential Equations / Edition 1

Foundations of the Classical Theory of Partial Differential Equations / Edition 1

by Yu.V. Egorov, R. Cooke, M.A. Shubin


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Product Details

ISBN-13: 9783540638254
Publisher: Springer Berlin Heidelberg
Publication date: 05/02/2001
Edition description: 1998
Pages: 259
Product dimensions: 6.10(w) x 9.25(h) x 0.24(d)

Table of Contents

1. Basic Concepts.- 1. Basic Definitions and Examples.- 1.1. The Definition of a Linear Partial Differential Equation.- 1.2. The Role of Partial Differential Equations in the Mathematical Modeling of Physical Processes.- 1.3. Derivation of the Equation for the Longitudinal Elastic Vibrations of a Rod.- 1.4. Derivation of the Equation of Heat Conduction.- 1.5. The Limits of Applicability of Mathematical Models.- 1.6. Initial and Boundary Conditions.- 1.7. Examples of Linear Partial Differential Equations.- 1.8. The Concept of Well-Posedness of a Boundary-value Problem. The Cauchy Problem.- 2. The Cauchy-Kovalevskaya Theorem and Its Generalizations.- 2.1. The Cauchy-Kovalevskaya Theorem.- 2.2. An Example of Nonexistence of an Analytic Solution.- 2.3. Some Generalizations of the Cauchy-Kovalevskaya Theorem. Characteristics.- 2.4. Ovsyannikov’s Theorem.- 2.5. Holmgren’s Theorem.- 3. Classification of Linear Differential Equations. Reduction to Canonical Form and Characteristics.- 3.1. Classification of Second-Order Equations and Their Reduction to Canonical Form at a Point.- 3.2. Characteristics of Second-Order Equations and Reduction to Canonical Form of Second-Order Equations with Two Independent Variables.- 3.3. Ellipticity, Hyperbolicity, and Parabolicity for General Linear Differential Equations and Systems.- 3.4. Characteristics as Solutions of the Hamilton-Jacobi Equation.- 2. The Classical Theory.- 1. Distributions and Equations with Constant Coefficients.- 1.1. The Concept of a Distribution.- 1.2. The Spaces of Test Functions and Distributions.- 1.3. The Topology in the Space of Distributions.- 1.4. The Support of a Distribution. The General Form of Distributions.- 1.5. Differentiation of Distributions.- 1.6. Multiplication of a Distribution by a Smooth Function. Linear Differential Operators in Spaces of Distributions.- 1.7. Change of Variables and Homogeneous Distributions.- 1.8. The Direct or Tensor Product of Distributions.- 1.9. The Convolution of Distributions.- 1.10. The Fourier Transform of Tempered Distributions.- 1.11. The Schwartz Kernel of a Linear Operator.- 1.12. Fundamental Solutions for Operators with Constant Coefficients.- 1.13. A Fundamental Solution for the Cauchy Problem.- 1.14. Fundamental Solutions and Solutions of Inhomogeneous Equations.- 1.15. Duhamel’s Principle for Equations with Constant Coefficients.- 1.16. The Fundamental Solution and the Behavior of Solutions at Infinity.- 1.17. Local Properties of Solutions of Homogeneous Equations with Constant Coefficients. Hypoellipticity and Ellipticity.- 1.18. Liouville’s Theorem for Equations with Constant Coefficients.- 1.19. Isolated Singularities of Solutions of Hypoelliptic Equations.- 2. Elliptic Equations and Boundary-Value Problems.- 2.1. The Definition of Ellipticity. The Laplace and Poisson Equations.- 2.2. A Fundamental Solution for the Laplacian Operator. Green’s Formula.- 2.3. Mean-Value Theorems for Harmonic Functions.- 2.4. The Maximum Principle for Harmonic Functions and the Normal Derivative Lemma.- 2.5. Uniqueness of the Classical Solutions of the Dirichlet and Neumann Problems for Laplace’s Equation.- 2.6. Internal A Priori Estimates for Harmonic Functions. Harnack’s Theorem.- 2.7. The Green’s Function of the Dirichlet Problem for Laplace’s Equation.- 2.8. The Green’s Function and the Solution of the Dirichlet Problem for a Ball and a Half-Space. The Reflection Principle.- 2.9. Harnack’s Inequality and Liouville’s Theorem.- 2.10. The Removable Singularities Theorem.- 2.11. The Kelvin Transform and the Statement of Exterior Boundary-Value Problems for Laplace’s Equation.- 2.12. Potentials.- 2.13. Application of Potentials to the Solution of Boundary-Value Problems.- 2.14. Boundary-Value Problems for Poisson’s Equation in Hölder Spaces. Schauder Estimates.- 2.15. Capacity.- 2.16. The Dirichlet Problem in the Case of Arbitrary Regions (The Method of Balayage). Regularity of a Boundary Point. The Wiener Regularity Criterion.- 2.17. General Second-Order Elliptic Equations. Eigenvalues and Eigenfunctions of Elliptic Operators.- 2.18. Higher-Order Elliptic Equations and General Elliptic Boundary-Value Problems. The Shapiro-Lopatinskij Condition.- 2.19. The Index of an Elliptic Boundary-Value Problem.- 2.20. Ellipticity with a Parameter and Unique Solvability of Elliptic Boundary-Value Problems.- 3. Sobolev Spaces and Generalized Solutions of Boundary-Value Problems.- 3.1. The Fundamental Spaces.- 3.2. Imbedding and Trace Theorems.- 3.3. Generalized Solutions of Elliptic Boundary-Value Problems and Eigenvalue Problems.- 3.4. Generalized Solutions of Parabolic Boundary-Value Problems.- 3.5. Generalized Solutions of Hyperbolic Boundary-Value Problems.- 4. Hyperbolic Equations.- 4.1. Definitions and Examples.- 4.2. Hyperbolicity and Well-Posedness of the Cauchy Problem.- 4.3. Energy Estimates.- 4.4. The Speed of Propagation of Disturbances.- 4.5. Solution of the Cauchy Problem for the Wave Equation.- 4.6. Huyghens’ Principle.- 4.7. The Plane Wave Method.- 4.8. The Solution of the Cauchy Problem in the Plane.- 4.9. Lacunae.- 4.10. The Cauchy Problem for a Strictly Hyperbolic System with Rapidly Oscillating Initial Data.- 4.11. Discontinuous Solutions of Hyperbolic Equations.- 4.12. Symmetric Hyperbolic Operators.- 4.13. The Mixed Boundary-Value Problem.- 4.14. The Method of Separation of Variables.- 5. Parabolic Equations.- 5.1. Definitions and Examples.- 5.2. The Maximum Principle and Its Consequences.- 5.3. Integral Estimates.- 5.4. Estimates in Hölder Spaces.- 5.5. The Regularity of Solutions of a Second-Order Parabolic Equation.- 5.6. Poisson’s Formula.- 5.7. A Fundamental Solution of the Cauchy Problem for a Second-Order Equation with Variable Coefficients.- 5.8. Shilov-Parabolic Systems.- 5.9. Systems with Variable Coefficients.- 5.10. The Mixed Boundary-Value Problem.- 5.11. Stabilization of the Solutions of the Mixed Boundary-Value Problem and the Cauchy Problem.- 6. General Evolution Equations.- 6.1. The Cauchy Problem. The Hadamard and Petrovskij Conditions.- 6.2. Application of the Laplace Transform.- 6.3. Application of the Theory of Semigroups.- 6.4. Some Examples.- 7. Exterior Boundary-Value Problems and Scattering Theory.- 7.1. Radiation Conditions.- 7.2. The Principle of Limiting Absorption and Limiting Amplitude.- 7.3. Radiation Conditions and the Principle of Limiting Absorption for Higher-Order Equations and Systems.- 7.4. Decay of the Local Energy.- 7.5. Scattering of Plane Waves.- 7.6. Spectral Analysis.- 7.7. The Scattering Operator and the Scattering Matrix.- 8. Spectral Theory of One-Dimensional Differential Operators.- 8.1. Outline of the Method of Separation of Variables.- 8.2. Regular Self-Adjoint Problems.- 8.3. Periodic and Antiperiodic Boundary Conditions.- 8.4. Asymptotics of the Eigenvalues and Eigenfunctions in the Regular Case.- 8.5. The Schrödinger Operator on a Half-Line.- 8.6. Essential Self-Adjointness and Self-Adjoint Extensions. The Weyl Circle and the Weyl Point.- 8.7. The Case of an Increasing Potential.- 8.8. The Case of a Rapidly Decaying Potential.- 8.9. The Schrödinger Operator on the Entire Line.- 8.10. The Hill Operator.- 9. Special Functions.- 9.1. Spherical Functions.- 9.2. The Legendre Polynomials.- 9.3. Cylindrical Functions.- 9.4. Properties of the Cylindrical Functions.- 9.5. Airy’s Equation.- 9.6. Some Other Classes of Functions.- References.- Author Index.

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