In this post, we will see the book Applications of Functional Analysis in Mathematical Physics – S. L. Sobolev.

# About the book

The present book arose as a result of revising a course of lectures given by the writer at the Leningrad State University. The notes for the lectures were taken and revised by H. L. Smolicki and I. A. Jakovlev, who contributed to them a series of valuable remarks and additions. Several additions, arising naturally during the lectures, were also made by the author himself.

In this fashion there came into being this monograph, a unifying treatment from a single point of view of a number of problems in the theory of partial differential equations. There are considered in it variational methods with applications to the Laplace equation and the polyharmonic equations as well as the Cauchy problem for linear and quasi-linear hyperbolic equations. The presentation of the problems of mathematical physics demands a suitable consideration of some new results and methods in functional analysis, which constitute in themselves the basis of all the later material. The first part is concerned with this basis. The material indicated above, the particular problems posed, and the methods for their investigation are not to be found in the ordinary course in mathematical physics and in particular, they are not in my book Equations of mathematical physics. The present book is of value for graduate students and research workers.

The book was translated from Russian by F. E. Browder was published in 1963.

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# Contents

## CHAPTER 1. SPECIAL PROBLEMS OF FUNCTIONAL ANALYSIS 1

§1. INTRODUCTION 1

1. Summable functions (1).

2. The Hélder and Minkowski inequalities (3).

3. The reverse of the Holder and Minkowski inequalities (7).

§2. BASIC PROPERTIES OF THE SPACES L_{p} 9

1. Norms. Definitions (9).

2. The Riesz-Fischer Theorem (11).

3. Continuity in the large of functions in L_{p} (11).

4. Countable dense nets (13).

§3. LINEAR FUNCTIONALS ON L_{p}

1. Definitions. Boundedness of linear functionals (16).

2. Clarkson’s inequalities (17).

3. Theorem on the general form of linear functionals (22).

4. Convergence of functionals (25).

$4. COMPACTNESS OF SPACES 28

1. Definition of compactness (28).

2. A theorem on weak compactness (29).

3. A theorem on strong compactness (30).

4. Proof of the theorem on strong compactness (31).

§5. GENERALIZED DERIVATIVES 33

1. Basic definitions (33).

2. Derivatives of averaged functions (35).

3. Rules for differentiation (37).

4. Independence of the domain (39).

§6. PROPERTIES OF INTEGRALS OF POTENTIAL TYPE 42

1. Integrals of potential type. Continuity (42).

2. Membership in L_{q} (43).

§7. THE SPACES L^{l}_{p} AND W^{l}_{p} 45

1. Definitions (45).

2. The norms in te (46).

3. Decompositions of wi and its norming (48).

4. Special decompositions of wi (50).

§8. IMBEDDING THEOREMS 56

1. The imbedding of W^{l}_{p} in C (56).

2. Imbedding of W^{l}_{p} in L_q (57).

3. Examples (58).

§9. GENERAL METHODS OF NORMING w? AND COROLLARIES OF THE IMBEDDING

THEOREM 60

1. A theorem on equivalent norms (60).

2. The general form of norms

equivalent to a given one (62).

3. Norms equivalent to the special norm (64).

4. Spherical projection operators (64).

5. Nonstar-like domains (66).

6. Examples (67).

§10. SOME CONSEQUENCES OF THE IMBEDDING THEOREM 68

1. Completeness of the space W,. (68).

2. The imbedding of We in Why (69).

3. Invariant norming of W¢) (72).

§11. THE COMPLETE CONTINUITY OF THE IMBEDDING OPERATOR (KONDRASEV’S

THEOREM) 74

1. Formulation of the problem (74). 2. A lemma on the compactness of the special integrals in C (75).

3. A lemma on the compactness of integrals in L_{q} (77).

4. Complete continuity of the imbedding operator in C (82).

5. Complete continuity of the operator of imbedding in L_{q} (84).

## CHAPTER II. VARIATIONAL METHODS IN MATHEMATICAL PHYSICS

§12. THE DIRICHLET PROBLEM 87

1. Introduction (87).

2. Solution of the variational problem (88). 3. Solution of the Dirichlet problem (91).

4. Uniqueness of the solution of the Dirichlet problem (94).

5. Hadamard’s example (97).

§13. THE NEUMANN PROBLEM 99

1. Formulation of the problem (99).

2. Solution of the variational problem (100).

3. Solution of the Neumann problem (101).

§14. POLYHARMONIC EQUATIONS 103

1. The behaviour of functions from W3$” on boundary manifolds of various dimensions (103).

2. Formulation of the basic boundary value problem (105).

3. Solution of the variational problem (106).

4. Solution of the basic boundary value problem (108).

§15. UNIQUENESS OF THE SOLUTION OF THE BASIC BOUNDARY VALUE PROBLEM

FOR THE POLYHARMONIC EQUATION 112

1. Formulation of the problem (112).

2. Lemma (112). 3. The structure of the domains 𝛺_{h} 𝛺_{3h} (115).

4. Proof of the lemma for k<|s/2| (116).

5. Proof of the lemma for k= [s/2]+1 (118).

6. Proof of the lemma for [s 2] + 2 ≤ k ≤ m (120).

7. Remarks on the formulation of the boundary conditions (122).

§16. THE EIGENVALUE PROBLEM 123

1. Introduction (123).

2. Auxiliary inequalities (124).

3. Minimal sequences and the equation of variations (126).

4. Existence of the first eigenfunction (128).

5. Existence of the later eigenfunctions (131).

6. The infinite sequence of eigenvalues (134).

7. Closedness of the set of eigenfunctions (136).

## CHAPTER III. THE THEORY OF HYPERBOLIC PARTIAL

DIFFERENTIAL EQUATIONS

§17. SOLUTION OF THE WAVE EQUATION WITH SMOOTH INITIAL CONDITIONS 139

1. Derivation of the basic inequality (139).

2. Estimates for the growth of the solution and its derivatives (142).

3. Solutions for special initial data (144).

4. Proof of the basic theorem (146).

§18. THE GENERALIZED CAUCHY PROBLEM FOR THE WAVE EQUATION 148

1. Twice differentiable solutions (148).

2. Example (150).

3. Generalized solutions (152).

4. Existence of initial data (153). 5. Solutions of the generalized Cauchy problem (155).

§19. LINEAR EQUATIONS OF NORMAL HYPERBOLIC TYPE WITH VARIABLE COEF-

FICIENTS (BASIC PROPERTIES) 157

1. Characteristics and bicharacteristics (157).

2. The characteristic conoid (164).

3. Equations in canonical coordinates (166).

4. The basic operators M^{(0)} and L^{(0)} in polar coordinates (168).

5. The system of basic relations on the cone (178).

§20. THE CAUCHY PROBLEM FOR LINEAR EQUATIONS WITH SMOOTH COEFFICIENTS 175

1. The operators adjoint to the operators of the basic system (175). 2. The construction of the functions 𝜎 (177).

3. Investigation of the properties of the functions 𝜎 (179). 4. Derivation of the basic integral identity Bu=SF (181).

5. The inverse integral operator B^{-1} and the method of successive

approximations (183).

6. The adjoint integral operator B* (187).

7. The adjoint integral operator S* (191).

8. Solution of the Cauchy problem for an even number of variables (192).

9. The Cauchy problem for an odd number of variables (195).

§21. INVESTIGATION OF LINEAR HYPERBOLIC EQUATIONS WITH VARIABLE COEFFICIENTS 196

1. Simplification of the equation (196).

2. Formulation of the Cauchy problem for generalized solutions (198).

3. Basic inequalities (200).

4. A lemma on estimates for approximating solutions (204).

5. Solution of the generalized problem (209).

6. Formulation of the classical Cauchy problem (210).

7. A lemma on estimates for derivatives (213).

8. Solution of the classical Cauchy problem (215).

§22. QUASILINEAR EQUATIONS 217

1. Formulation of functions of functions (217).

2. Basic inequalities (221).

3. Petrovsky’s functional equation (224).

4. The Cauchy problem with homogeneous initial conditions (229).

5. Properties of averaged functions (231).

6. Transformation of initial conditions (235).

7. The general case for the Cauchy problem for quasilinear equations (237).

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