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.- 1.3 Minimising Special Classes of Functions.- 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.- 2.3 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.- 5.3 Symmetries and Conservation Laws in Classical Mechanics.- 5.4 The Brachystochrone Problem.- 5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory.- 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.- 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.- 8.3 Ljusternik-Schnirelman Theory for Compact Manifolds.- 8.4 The Existence of Infinitely Many Solutions of Nonlinear Elliptic Eigenvalue Problems.- 9. Semilinear Elliptic Differential Equations. Some Recent Results on Global Solutions.- 9.1 Introduction.- 9.2 Technical Preliminaries.- 9.3 Some Properties of Weak Solutions of Semilinear Elliptic Equations.- 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.- 9.8 Existence of a Minimiser II: Some Examples.- 9.9 Nonlinear Field Equations in Two Dimensions.- 9.10 Conclusion and 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$$Vleft( x right) = Sigma _{j = 1}^kfrac{{{z_j}}}{{left| {x - {x_j}} right|}}$$.- 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.