Variational Methods in Mathematical Physics: A Unified Approach (Theoretical and Mathematical Physics)

Title......Page 1
Copyright Page......Page 2
Preface......Page 3
Contents......Page 7
Some Remarks on the History and Objectives of the Calculus of Variations......Page 11
1.1 The Fundamental Theorem of the Calculus of Variations......Page 25
1.2 Applying the Fundamental Theorem in Banach Spaces......Page 30
1.2.1 Sequentially Lower Semicontinuous Functionals......Page 32
1.3 Minimising Special Classes of Functions......Page 35
1.3.1 Quadratic Functionals......Page 38
1.4 Some Remarks on Linear Optimisation......Page 40
1.5 Ritz's Approximation Method......Page 41
2.1 General Remarks......Page 45
2.2 The Frechet Derivative......Page 46
2.2.1 Higher Derivatives......Page 53
2.2.2 Some Properties of Frechet Derivatives......Page 54
2.3 The Gateaux Derivative......Page 56
2.4 nth Variation......Page 59
2.5 The Assumptions of the Fundamental Theorem  of Variational Calculus......Page 61
2.6 Convexity of f and Monotonicity of f'......Page 62
3.1 Extrema and Critical Values......Page 64
3.2 Necessary Conditions for an Extremum......Page 65
3.3 Sufficient Conditions for an Extremum......Page 70
4.1 Geometrical Interpretation  of Constrained Minimisation Problems......Page 73
4.2 Ljusternik's Theorems......Page 76
4.3 Necessary and Sufficient Conditions  for Extrema Subject to Constraints......Page 82
4.4 A Special Case......Page 85
5.1 General Remarks......Page 87
5.2 Hamilton's Principle in Classical Mechanics......Page 90
5.2.1 Systems with One Degree of Freedom......Page 91
5.2.2 Systems with Several Degrees of Freedom......Page 105
5.2.3 An Example from Classical Mechanics......Page 115
5.3.1 Hamiltonian Formulation of Classical Mechanics......Page 117
5.3.2 Coordinate Transformations and Integrals of Motion......Page 119
5.4 The Brachystochrone Problem......Page 123
5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory......Page 126
5.5.1 Hamilton's Principle in Local Field Theory......Page 127
5.5.2 Examples of Local Classical Field Theories......Page 132
5.6 Noether's Theorem in Classical Field Theory......Page 134
5.7 The Principle of Symmetric Criticality......Page 140
6.1 The Spectral Theorem for Compact Self-Adjoint Operators.  Courant's Classical Minimax Principle. Projection Theorem......Page 152
6.2 Differential Operators and Forms......Page 158
6.3 The Theorem of Lax-Milgram and Some Generalisations......Page 162
6.4 The Spectrum of Elliptic Differential Operators in a Bounded  Domain. Some Problems from Classical Potential Theory......Page 166
6.5 Variational Solution of Parabolic Differential Equations. The Heat Conduction Equation. The Stokes Equations......Page 169
6.5.1 A General Framework for the Variational Solution of Parabolic Problems......Page 171
6.5.2 The Heat Conduction Equation......Page 176
6.5.3 The Stokes Equations in Hydrodynamics......Page 177
7.1 Forms and Operators - Boundary Value Problems......Page 181
7.2 Surjectivity of Coercive Monotonic Operators.  Theorems of Browder and Minty......Page 183
7.3 Nonlinear Elliptic Boundary Value Problems.  A Variational Solution......Page 188
8.1 Introduction......Page 202
8.2.1 Abstract Versions of Some Existence Theorems......Page 205
8.2.2 Determining the Ground State Solution  for Nonlinear Elliptic Eigenvalue Problems......Page 213
8.3.1 The Topological Basis of the Generalised Minimax Principle......Page 215
8.3.2 The Deformation Theorem......Page 217
8.3.3 The Ljusternik-Schnirelman Category and the Genus of a Set......Page 220
8.3.4 Minimax Characterisation of Critical Values  of Ljusternik-Schnirelman......Page 225
8.4.1 Sphere-Like Constraints......Page 227
8.4.2 Galerkin Approximation for Nonlinear Eigenvalue Problems  in Separable Banach Spaces......Page 230
8.4.3 The Existence of Infinitely Many Critical Points as Solutions of Abstract Eigenvalue Problems in Separable Banach Spaces......Page 235
8.4.4 The Existence of Infinitely Many Solutions of Nonlinear Eigenvalue Problems......Page 238
9.1 Introduction......Page 251
9.2.1 Some Function Spaces and Their Properties......Page 257
9.2.2 Some Continuity Results for Niemytski Operators......Page 262
9.2.3 Some Results on Concentration of Function Sequences......Page 266
9.2.4. A One-dimensional Variational Problem......Page 272
9.3.1 Regularity of Weak Solutions......Page 276
9.3.2 Pohozaev's Identities......Page 288
9.4 Best Constant in Sobolev Inequality......Page 293
9.5 The Local Case with Critical Sobolev Exponent......Page 297
9.6 The Constrained Minimisation Method Under Scale Covariance......Page 304
9.7.1 Symmetries......Page 312
9.7.2. Necessary and Sufficient Conditions......Page 314
9.7.3 The Concentration Condition......Page 315
9.7.4 Minimising Subsets......Page 318
9.7.5 Growth Restrictions on the Potential......Page 320
9.8 Existence of a Minimiser II: Some Examples......Page 322
9.8.1 Some Non-translation-invariant Cases......Page 323
9.8.2 Spherically Symmetric Cases......Page 326
9.8.3 The Translation-invariant Case Without Spherical Symmetry......Page 329
9.9 Nonlinear Field Equations in Two Dimensions......Page 332
9.9.1 Some Properties of Niemytski Operators on E.......Page 333
9.9.2 Solution of Some Two-Dimensional Vector Field Equations......Page 336
9.10.1 Conclusion......Page 342
9.10.2 Generalisations......Page 344
9.10.3 Comments......Page 345
9.11 Complementary Remarks......Page 347
10.1 General Remarks......Page 350
10.2 Some Results from the Theory of LP Spaces (1 < p < oo)......Page 352
10.3 Minimisation of the Thomas-Fermi Energy Functional......Page 354
10.4 Thomas-Fermi Equations and the Minimisation Problem  for the TF Functional......Page 361
10.5 Solution of TF Equations for Potentials......Page 367
10.6 Remarks on Recent Developments in Thomas-Fermi and Related Theories......Page 371
Appendix A. Banach Spaces......Page 373
Appendix B. Continuity and Semicontinuity......Page 381
Appendix C. Compactness in Banach Spaces......Page 383
D.1 Definition and Properties......Page 390
D.2 Poincare's Inequality......Page 395
D.3 Continuous Embeddings of Sobolev Spaces......Page 396
D.4 Compact Embeddings of Sobolev Spaces......Page 398
E.1 Bessel Potentials......Page 401
E.2 Some Properties of Weakly Differentiable Functions......Page 402
E.3 Proof of Theorem 9.2.3......Page 403
References......Page 405
Index of Names......Page 415
Subject Index......Page 417