In this post, we will see the book Partial Differential Equations of Mathematical Physics – S. L. Sobolev.

# About the book

The classical partial differential equations of mathematical physics, formulated and intensively studied by the great mathematicians of the nineteenth century, remain the foundation of investigations into waves, heat conduction, hydrodynamics, and other physical problems. These equations, in the early twentieth century, prompted further mathematical researches, and in turn themselves benefited by the application of new methods in pure mathematics. The theories of sets and of Lebesgue integration enable us to state conditions and to characterize solutions in a much more precise fashion; a differential equation with the boundary conditions to be imposed on its solution can be absorbed into a single formulation as an integral equation; Green’s function permits a formal explicit solution; eigenvalues and eigenfunctions generalize Fourier’s analysis to a wide variety of problems.

All these matters are dealt with in Sobolev’s book, without assumption of previous acquaintance. The reader has only to be familiar with elementary analysis; from there he is introduced to these more advanced concepts, which are developed in detail and with great precision as far as they are required for the main purposes of the book. Care has been taken to render the exposition suitable for a novice in this field: theorems are often approached through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems.

The book was translated from Russian by E. R. Dawson was published in 1964.

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# Contents

TRANSLATION EDITOR’S PREFACE ix

AUTHOR’S PREFACES TO THE FIRST AND THIRD EDITIONS x

## LECTURE 1. DERIVATION OF THE FUNDAMENTAL EQUATIONS 1

§ 1. Ostrogradski’s Formula 1

§ 2. Equation for Vibrations of a String 3

§ 3. Equation for Vibrations of a Membrane 6

§ 4. Equation of Continuity for Motion of a Fluid. Laplace’s Equation 8

§ 5. Equation of Heat Conduction 13

§ 6. Sound Waves 17

## LECTURE 2. THE FORMULATION OF PROBLEMS OF MATHEMATICAL PHYSICS.

HADAMARD’S EXAMPLE 22

§ 1. Initial Conditions and Boundary Conditions 22

§ 2. The Dependence of the Solution on the Boundary Conditions. Hadamard’s Example 26

## LECTURE 3. THE CLASSIFICATION OF LINEAR EQUATIONS OF THE SECOND ORDER 33

§ 1. Linear Equations and Quadratic Forms. Canonical Form of an Equation 33

§ 2. Canonical Form of Equations in Two Independent Variables 38

§ 3. Second Canonical Form of Hyperbolic Equations in Two Independent Variables 42

§ 4. Characteristics 43

## LECTURE 4. THE EQUATION FOR A VIBRATING STRING AND ITS SOLUTION BY D’ALEMBERT’S METHOD 46

§ 1. D’Alembert’s Formula. Infinite String 46

§ 2. String with Two Fixed Ends 49

§ 3. Solution of the Problem for a Non-Homogeneous Equation and for More General Boundary Conditions 51

## LECTURE 5. RIEMANN’S METHOD 58

§ 1. The Boundary-Value Problem of the First Kind for Hyperbolic Equations 58

§ 2. Adjoint Differential Operators 62

§ 3. Riemann’s Method 65

§ 4. Riemann’s Function for the Adjoint Equation 68

§ 5. Some Qualitative Consequences of Riemann’s Formula 71

## LECTURE 6. MULTIPLE INTEGRALS: LEBESGUE INTEGRATION 72

§ 1. Closed and Open Sets of Points 73

§ 2. Integrals of Continuous Functions on Open Sets 79

§ 3. Integrals of Continuous Functions on Bounded Closed Sets 85

§ 4. Summable Functions 92

§ 5. The Indefinite Integral of a Function of One Variable. Examples 99

§ 6. Measurable Sets. Egorov’s Theorem 103

§ 7. Convergence in the Mean of Summable Functions 111

§ 8. The Lebesgue—Fubini Theorem 121

## LECTURE 7. INTEGRALS DEPENDENT ON A PARAMETER 126

§ 1. Integrals which are Uniformly Convergent for a Given Value of Parameter 126

§ 2. The Derivative of an Improper Integral with respect to a Parameter 129

## LECTURE 8. THE EQUATION OF HEAT CONDUCTION 133

§ 1. Principal Solution 133

§ 2. The Solution of Cauchy’s Problem 139

## LECTURE 9. LAPLACE’S EQUATION AND POISSON’s EQUATION 146

§ 1. The Theorem of the Maximum 146

§ 2. The Principal Solution. Green’s Formula 148

§ 3. The Potential due to a Volume, to a Single Layer, and to a Double Layer 150

## LECTURE 10. SOME GENERAL CONSEQUENCES OF GREEN’S FORMULA 155

§ 1. The Mean-Value Theorem for a Harmonic Function 155

§ 2. Behaviour of a Harmonic Function near a Singular Point 158

§ 3. Behaviour of a Harmonic Function at Infinity. Inverse Points 162

## LECTURE 11. POISSON’S EQUATION IN AN UNBOUNDED MEDIUM. NEWTONIAN

POTENTIAL 166

## LECTURE 12. THE SOLUTION OF THE DIRICHLET PROBLEM FOR A SPHERE 171

## LECTURE 13. THE DIRICHLET PROBLEM AND THE NEUMANN PROBLEM FOR A

HALF-SPACE 180

## LECTURE 14. THE WAVE EQUATION AND THE RETARDED POTENTIAL 188

§ 1. The Characteristics of the Wave Equation 188

§ 2. Kirchhoff’s Method of Solution of Cauchy’s Problem 189

## LECTURE 15. PROPERTIES OF THE POTENTIALS OF SINGLE AND DOUBLE LAYERS 202

§ 1. General Remarks 202

§ 2. Properties of the Potential of a Double Layer 203

§ 3. Properties of the Potential of a Single Layer 210

§ 4. Regular Normal Derivative 217

§ 5. Normal Derivative of the Potential of a Double Layer 218

§ 6. Behaviour of the Potentials at Infinity 220

## LECTURE 16. REDUCTION OF THE DIRICHLET PROBLEM AND THE NEUMANN

PROBLEM TO INTEGRAL EQUATIONS 222

§ 1. Formulation of the Problems and the Uniqueness of their Solutions 222

§ 2. The Integral Equations for the Formulated Problems 225

## LECTURE 17. LAPLACE’S EQUATION AND POISSON’S EQUATION IN A PLANE 228

§ 1. The Principal Solution 228

§ 2. The Basic Problems 230

§ 3. The Logarithmic Potential 234

## LECTURE 18. THE THEORY OF INTEGRAL EQUATIONS 237

§ 1. General Remarks 237

§ 2. The Method of Successive Approximations 238

§ 3. Volterra Equations 242

§ 4. Equations with Degenerate Kernel 243

§ 5. A Kernel of Special Type. Fredholm’s Theorems 248

§ 6. Generalization of the Results 253

§ 7. Equations with Unbounded Kernels of a Special Form 256

## LECTURE 19. APPLICATION OF THE THEORY OF FREDHOLM EQUATIONS TO THE

SOLUTION OF THE DIRICHLET AND NEUMANN PROBLEMS 258

§ 1. Derivation of the Properties of Integral Equations 258

§ 2. Investigation of the Equations 260

## LECTURE 20. GREEN’S FUNCTION 265

§ 1. The Differential Operator with One Independent Variable 265

§ 2. Adjoint Operators and Adjoint Families 268

§ 3. The Fundamental Lemma on the Integrals of Adjoint Equations 271

§ 4. The Influence Function 275

§ 5. Definition and Construction of Green’s Function 278

§ 6. The Generalized Green’s Function for a Linear Second-Order Equation 281

§ 7. Examples 286

## LECTURE 21. GREEN’S FUNCTION FOR THE LAPLACE OPERATOR 291

§ 1. Green’s Function for the Dirichlet Problem 291

§ 2. The Concept of Green’s Function for the Neumann Problem 296

## LECTURE 22. CORRECTNESS OF FORMULATION OF THE BOUNDARY-VALUE

PROBLEMS OF MATHEMATICAL PHYSICS 301

§ 1. The Equation of Heat Conduction 301

§ 2. The Concept of the Generalized Solution 304

§ 3. The Wave Equation 307

§ 4. The Generalized Solution of the Wave Equation 311

§ 5. A Property of Generalized Solutions of Homogeneous Equations 317

§ 6. Bunyakovski’s Inequality and Minkovski’s Inequality 322

§ 7. The Riesz—Fischer Theorem 324

## LECTURE 23. FOURIER’S METHOD 327

§ 1. Separation of the Variables 327

§ 2. The Analogy between the Problems of Vibrations of a Continuous Medium and Vibrations of Mechanical Systems with a Finite Number of Degrees of Freedom 334

§ 3. The Inhomogeneous Equation 336

§ 4. Longitudinal Vibrations of a Bar 339

## LECTURE 24. INTEGRAL EQUATIONS WITH REAL, SYMMETRIC KERNELS 342

§ 1. Elementary Properties. Completely Continuous Operators 342

§ 2. Proof of the Existence of an Eigenvalue 354

## LECTURE 25. THE BILINEAR FORMULA AND THE HILBERT—SCHMIDT THEOREM 357

§ 1. The Bilinear Formula 357

§ 2. The Hilbert-Schmidt Theorem 364

§ 3. Proof of the Fourier Method for the Solution of the Boundary-Value Problems of Mathematical Physics 367

§ 4. An Application of the Theory of Integral Equations with Symmetric Kernel 375

## LECTURE 26. THE INHOMOGENEOUS INTEGRAL EQUATION WITH A SYMMETRIC

KERNEL 376

§ 1. Expansion of the Resolvent 376

§ 2. Representation of the Solution by means of Analytical Functions 378

## LECTURE 27. VIBRATIONS OF A RECTANGULAR PARALLELEPIPED 382

## LECTURE 28. LAPLACE’S EQUATION IN CURVILINEAR COORDINATES. EXAMPLES OF THE USE OF FOURIER’S METHOD 388

§ 1. Laplace’s Equation in Curvilinear Coordinates 388

§ 2. Bessel Functions 394

§ 3. Complete Separation of the Variables in the Equation V? u = 0 in Polar Coordinates 397

## LECTURE 29. HARMONIC POLYNOMIALS AND SPHERICAL FUNCTIONS 401

§ 1. Definition of Spherical Functions 401

§ 2. Approximation by means of Spherical Harmonics 404

§ 3. The Dirichlet Problem for a Sphere 407

§ 4. The Differential Equations for Spherical Functions 408

## LECTURE 30. SOME ELEMENTARY PROPERTIES OF SPHERICAL FUNCTIONS 414

§ 1. Legendre Polynomials 414

§ 2. The Generating Function 415

§ 3. Laplace’s Formula 418

INDEX 421

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