Variational Methods in Mathematical Physics

Some Remarks on the History and Objectives of the Calculus of Variations.- 1. Direct Methods of the Calculus of Variations.- 1.1 The Fundamental Theorem of the Calculus of Variations.- 1.2 Applying the Fundamental Theorem in Banach Spaces.- 1.2.1 Sequentially Lower Semicontinuous Functionals.- 1.3 Minimising Special Classes of Functions.- 1.3.1 Quadratic Functionals.- 1.4 Some Remarks on Linear Optimisation.- 1.5 Ritz's Approximation Method.- 2. Differential Calculus in Banach Spaces.- 2.1 General Remarks.- 2.2 The Fréchet Derivative.- 2.2.1 Higher Derivatives.- 2.2.2 Some Properties of Fréchet Derivatives.- 2.3 The Gâteaux Derivative.- 2.4 nth Variation.- 2.5 The Assumptions of the Fundamental Theorem of Variational Calculus.- 2.6 Convexity of f and Monotonicity of f ?.- 3. Extrema of Differentiable Functions.- 3.1 Extrema and Critical Values.- 3.2 Necessary Conditions for an Extremum.- 3.3 Sufficient Conditions for an Extremum.- 4. Constrained Minimisation Problems (Method of Lagrange Multipliers).- 4.1 Geometrical Interpretation of Constrained Minimisation Problems.- 4.2 Ljusternik's Theorems.- 4.3 Necessary and Sufficient Conditions for Extrema Subject to Constraints.- 4.4 A Special Case.- 5. Classical Variational Problems.- 5.1 General Remarks.- 5.2 Hamilton's Principle in Classical Mechanics.- 5.2.1 Systems with One Degree of Freedom.- 5.2.2 Systems with Several Degrees of Freedom.- 5.2.3 An Example from Classical Mechanics.- 5.3 Symmetries and Conservation Laws in Classical Mechanics.- 5.3.1 Hamiltonian Formulation of Classical Mechanics.- 5.3.2 Coordinate Transformations and Integrals of Motion.- 5.4 The Brachystochrone Problem.- 5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory.- 5.5.1 Hamilton's Principle in Local Field Theory.- 5.5.2 Examples of Local Classical Field Theories.- 5.6 Noether's Theorem in Classical Field Theory.- 5.7 The Principle of Symmetric Criticality.- 6. The Variational Approach to Linear Boundary and Eigenvalue Problems.- 6.1 The Spectral Theorem for Compact Self-Adjoint Operators. Courant's Classical Minimax Principle. Projection Theorem.- 6.2 Differential Operators and Forms.- 6.3 The Theorem of Lax-Milgram and Some Generalisations.- 6.4 The Spectrum of Elliptic Differential Operators in a Bounded Domain. Some Problems from Classical Potential Theory.- 6.5 Variational Solution of Parabolic Differential Equations. The Heat Conduction Equation. The Stokes Equations.- 6.5.1 A General Framework for the Variational Solution of Parabolic Problems.- 6.5.2 The Heat Conduction Equation.- 6.5.3 The Stokes Equations in Hydrodynamics.- 7. Nonlinear Elliptic Boundary Value Problems and Monotonic Operators.- 7.1 Forms and Operators - Boundary Value Problems.- 7.2 Surjectivity of Coercive Monotonic Operators. Theorems of Browder and Minty.- 7.3 Nonlinear Elliptic Boundary Value Problems. A Variational Solution.- 8. Nonlinear Elliptic Eigenvalue Problems.- 8.1 Introduction.- 8.2 Determination of the Ground State in Nonlinear Elliptic Eigenvalue Problems.- 8.2.1 Abstract Versions of Some Existence Theorems.- 8.2.2 Determining the Ground State Solution for Nonlinear Elliptic Eigenvalue Problems.- 8.3 Ljusternik-Schnirelman Theory for Compact Manifolds.- 8.3.1 The Topological Basis of the Generalised Minimax Principle.- 8.3.2 The Deformation Theorem.- 8.3.3 The Ljusternik-Schnirelman Category and the Genus of a Set.- 8.3.4 Minimax Characterisation of Critical Values of Ljusternik-Schnirelman.- 8.4 The Existence of Infinitely Many Solutions of Nonlinear Elliptic Eigenvalue Problems.- 8.4.1 Sphere-Like Constraints.- 8.4.2 Galerkin Approximation for Nonlinear Eigenvalue Problems in Separable Banach Spaces.- 8.4.3 The Existence of Infinitely Many Critical Points as Solutions of Abstract Eigenvalue Problems in Separable Banach Spaces.- 8.4.4 The Existence of Infinitely Many Solutions of Nonlinear Eigenvalue Problems.- 9. Semilinear Elliptic Differential Equations. Some Recent Results on Global Solutions.- 9.1 Introduction.- 9.2 Technical Preliminaries.- 9.2.1 Some Function Spaces and Their Properties.- 9.2.2 Some Continuity Results for Niemytski Operators.- 9.2.3 Some Results on Concentration of Function Sequences.- 9.2.4. A One-dimensional Variational Problem.- 9.3 Some Properties of Weak Solutions of Semilinear Elliptic Equations.- 9.3.1 Regularity of Weak Solutions.- 9.3.2 Pohozaev's Identities.- 9.4 Best Constant in Sobolev Inequality.- 9.5 The Local Case with Critical Sobolev Exponent.- 9.6 The Constrained Minimisation Method Under Scale Covariance.- 9.7 Existence of a Minimiser I: Some General Results.- 9.7.1 Symmetries.- 9.7.2. Necessary and Sufficient Conditions.- 9.7.3 The Concentration Condition.- 9.7.4 Minimising Subsets.- 9.7.5 Growth Restrictions on the Potential.- 9.8 Existence of a Minimiser II: Some Examples.- 9.8.1 Some Non-translation-invariant Cases.- 9.8.2 Spherically Symmetric Cases.- 9.8.3 The Translation-invariant Case Without Spherical Symmetry.- 9.9 Nonlinear Field Equations in Two Dimensions.- 9.9.1 Some Properties of Niemytski Operators on Eq.- 9.9.2 Solution of Some Two-Dimensional Vector Field Equations.- 9.10 Conclusion and Comments.- 9.10.1 Conclusion.- 9.10.2 Generalisations.- 9.10.3 Comments.- 9.11 Complementary Remarks.- 10. Thomas-Fermi Theory.- 10.1 General Remarks.- 10.2 Some Results from the Theory of Lp Spaces (1 ? p ? ?).- 10.3 Minimisation of the Thomas-Fermi Energy Functional.- 10.4 Thomas-Fermi Equations and the Minimisation Problem for the TF Functional.- 10.5 Solution of TF Equations for Potentials of the Form$$V\left( x \right) = \Sigma _{j = 1}^k\frac{{{z_j}}}{{\left| {x - {x_j}} \right|}}$$.- 10.6 Remarks on Recent Developments in Thomas-Fermi and Related Theories.- Appendix A. Banach Spaces.- Appendix B. Continuity and Semicontinuity.- Appendix C. Compactness in Banach Spaces.- D.1 Definition and Properties.- D.2 Poincaré's Inequality.- D.3 Continuous Embeddings of Sobolev Spaces.- D.4 Compact Embeddings of Sobolev Spaces.- Appendix E.- E.1 Bessel Potentials.- E.2 Some Properties of Weakly Differentiable Functions.- E.3 Proof of Theorem 9.2.3.- References.- Index of Names.