Some Applications of Functional Analysis in Mathematical Physics (Translations of Mathematical Monographs)

This book presents the theory of functions spaces, now known as Sobolev spaces, which are widely used in the theory of partial differential equations, mathematical physics, and numerous applications. The author also treats the variational method of solution of boundary value problems for elliptic equations, including those with boundary conditions given on manifolds of different dimensions. In addition, the theory of the Cauchy problem for second-order hyperbolic equations with variable coefficients is studied. The book is intended for researchers in mathematics and mathematical physics and would be useful to undergraduate and graduate students taking advanced courses in these areas. Cover......Page 1 Title......Page 2 Copyright......Page 3 Contents......Page 4 Preface to the Third Edition......Page 6 Preface to the First Edition......Page 8 1.1. Introduction......Page 10 1.2. Basic properties of the spaces L_P......Page 17 1.3. Linear functionals on L_P......Page 25 1.4. Compactness in L_P......Page 36 1.5. Generalized derivatives......Page 41 1.6. Properties of integrals of potential type......Page 48 1.7. The spaces L_p and W......Page 51 1.8. Imbedding theorems......Page 63 1.9. General methods of norming W_P(1) and corollaries of the imbedding theorems......Page 66 1.10. Some consequences of the imbedding theorems......Page 73 1.11. The complete continuity of the imbedding operator (Kondrashov's Theorem)......Page 79 2.1. The Dirichlet problem......Page 90 2.2. The Neumann problem......Page 101 2.3. Polyharmonic equations......Page 104 2.4. Uniqueness of the solution of the basic boundary value problem for the polyharmonic equation......Page 112 2.5. The eigenvalue problem......Page 124 3.1. Solution of the Cauchy problem for the wave equation with smooth initial conditions......Page 138 3.2. The generalized Cauchy problem for the wave equation......Page 146 3.3. Linear equations of normal hyperbolic type with variable coefficients (basic properties)......Page 155 3.4. The Cauchy problem for linear equations with smooth coefficients......Page 171 3.5. Investigation of linear hyperbolic equations with variable coefficients......Page 189 Comments......Page 208 Appendix: Methode nouvelle A resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales......Page 226 Comments on the Appendix......Page 262 Bibliography......Page 278 Index......Page 294 Presents the theory of functions spaces, known as Sobolev spaces, which are widely used in the theory of partial differential equations, mathematical physics, and numerous applications. This book also covers the variational method of solution of boundary value problems for elliptic equations.