Functional Analysis in Mechanics

L. P. Lebedev, Iosif I. Vorovich9780387955193, 0-387-95519-4

This book covers functional analysis and its applications to continuum mechanics. The mathematical material is treated in a non-abstract manner and is fully illuminated by the underlying mechanical ideas. The presentation is concise but complete, and is intended for specialists in continuum mechanics who wish to understand the mathematical underpinnings of the discipline. Graduate students and researchers in mathematics, physics, and engineering will find this book useful. Exercises and examples are included throughout with detailed solutions provided in the appendix.

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Table of contents :
Functional Analysis in Mechanics……Page 4
Preface to the English Edition……Page 6
Preface to the Russian Edition……Page 8
Contents……Page 10
Introduction……Page 14
1.1 Preliminaries……Page 20
1.2 Some Metric Spaces of Functions……Page 25
1.3 Energy Spaces……Page 27
1.5 Convergence in a Metric Space……Page 31
1.6 Completeness……Page 32
1.7 The Completion Theorem……Page 34
1.8 The Lebesgue Integral and the Space Lp(Ω)……Page 36
1.9 Banach and Hilbert Spaces……Page 40
1.10 Some Energy Spaces……Page 45
1.11 Sobolev Spaces……Page 60
1.12 Introduction to Operators……Page 63
1.13 Contraction Mapping Principle……Page 65
1.14 Generalized Solutions in Mechanics……Page 70
1.15 Separability……Page 75
1.16 Compactness, Hausdorff Criterion……Page 80
1.17 Arzela’s Theorem and Its Applications……Page 83
1.18 The Theory of Approximation in a Normed Space……Page 89
1.19 Decomposition Theorem, Riesz Representation……Page 92
1.20 Existence of Energy Solutions to Some Mechanics Problems……Page 96
1.21 The Problem of Elastico-Plasticity; Small Deformations……Page 100
1.22 Bases and Complete Systems……Page 107
1.23 Weak Convergence in a Hilbert Space……Page 112
1.24 The Ritz and Bubnov–Galerkin Methods in Linear Problems……Page 122
1.25 Curvilinear Coordinates, Non-Homogeneous Boundary Conditions……Page 124
1.26 The Bramble–Hilbert Lemma and Its Applications……Page 127
2.1 Spaces of Linear Operators……Page 134
2.2 Banach–Steinhaus Principle……Page 137
2.3 The Inverse Operator……Page 139
2.4 Closed Operators……Page 142
2.5 The Notion of Adjoint Operator……Page 145
2.6 Compact Operators……Page 152
2.7 Compact Operators in Hilbert Space……Page 157
2.8 Functions Taking Values in a Banach Space……Page 159
2.9 Spectrum of Linear Operators……Page 162
2.10 Resolvent Set of a Closed Linear Operator……Page 165
2.11 Spectrum of Compact Operators in Hilbert Space……Page 167
2.12 Analytic Nature of the Resolvent of a Compact Linear Operator……Page 175
2.13 Spectrum of Holomorphic Compact Operator Function……Page 177
2.14 Spectrum of Self-Adjoint Compact Linear Operator in Hilbert Space……Page 179
2.15 Some Applications of Spectral Theory……Page 184
2.16 Courant’s Minimax Principle……Page 188
3.1 Frechet and Gateaux Derivatives……Page 190
3.2 Liapunov–Schmidt Method……Page 195
3.3 Critical Points of a Functional……Page 197
3.4 Von Karman Equations of a Plate……Page 202
3.5 Buckling of a Thin Elastic Shell……Page 208
3.6 The Nonlinear Problem of Equilibrium of the Theory of Elastic Shallow Shells……Page 217
3.7 Degree Theory……Page 222
3.8 Steady-State Flow of Viscous Liquid……Page 224
Appendix Hints for Selected Problems……Page 232
References……Page 244
Index……Page 248