Variational Methods in Mathematical Physics: A Unified Approach

The first edition (in German) had the prevailing character of a textbook owing to the choice of material and the manner of its presentation. This second (translated, revised, and extended) edition, however, includes in its new parts considerably more recent and advanced results and thus goes partially beyond the textbook level. We should emphasize here that the primary intentions of this book are to provide (so far as possible given the restrictions of space) a selfcontained presentation of some modern developments in the direct methods of the cal­ culus of variations in applied mathematics and mathematical physics from a unified point of view and to link it to the traditional approach. These modern developments are, according to our background and interests: (i) Thomas-Fermi theory and related theories, and (ii) global systems of semilinear elliptic partial-differential equations and the existence of weak solutions and their regularity. Although the direct method in the calculus of variations can naturally be considered part of nonlinear functional analysis, we have not tried to present our material in this way. Some recent books on nonlinear functional analysis in this spirit are those by K. Deimling (Nonlinear Functional Analysis, Springer, Berlin Heidelberg 1985) and E. Zeidler (Nonlinear Functional Analysis and Its Applications, Vols. 1-4; Springer, New York 1986-1990).

Table of Contents:
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$$V\left( x \right) = \Sigma _{j = 1}^k\frac{{{z_j}}}{{\left| {x - {x_j}} \right|}}$$.- 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.