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