Variational Methods in Mathematical Physics

This is a nicely organized introduction to the direct methods of variational calculus focusing on the study of critical points of nonlinear functionals. It addresses students of mathematics and physics, and gives many applications to mechanics, classical field theory, boundary and eigenvalue problems of important partial differential equations, and also modern Thomas-Fermi theory.

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.