First variation

Abstract: 1 Introduction.- 1.1. Mean Curvature.- 1.2. Laplace's Equation.- 1.3. Angle of Contact.- 1.4. The Method of Gauss Characterization of the Energies.- 1.5. Variational Considerations.- 1.6. The Equation and the Boundary Condition.- 1.7. Divergence Structure.- 1.8. The Problem as a Geometrical One.- 1.9. The Capillary Tube.- 1.10. Dimensional Considerations.- Notes to Chapter 1.- 2 The Symmetric Capillary Tube.- 2.1. Historical and General.- 2.2. The Narrow Tube Center Height.- 2.3. The Narrow Tube Outer Height.- 2.4. The Narrow Tube Estimates Throughout the Trajectory.- 2.5. Height Estimates for Tubes of General Size.- 2.6. Meniscus Height Narrow Tubes.- 2.7. Meniscus Height General Case.- 2.8. Comparisons with Earlier Theories.- Notes to Chapter 2.- 3 The Symmetric Sessile Drop.- 3.1. The Correspondence Principle.- 3.2. Continuation Properties.- 3.3. Uniqueness and Existence.- 3.4. The Envelope.- 3.5. Comparison Theorems.- 3.6. Geometry of the Sessile Drop Small Drops.- 3.7. Geometry of the Sessile Drop Larger Drops.- Notes to Chapter 3.- 4 The Pendent Liquid Drop.- 4.1. Mise en Scene.- 4.2. Local Existence.- 4.3. Uniqueness.- 4.4. Global Behavior General Remarks.- 4.5. Small |u0|.- 4.6. Appearance of Vertical Points.- 4.7. Behavior for Large |u0|.- 4.8. Global Behavior.- 4.9. Maximum Vertical Diameter.- 4.10. Maximum Diameter.- 4.11. Maximum Volume.- 4.12. Asymptotic Properties.- 4.13. The Singular Solution.- 4.14. Isolated Character of Global Solutions.- 4.15. Stability.- Notes to Chapter 4.- 5 Asymmetric Case Comparison Principles and Applications.- 5.1. The General Comparison Principle.- 5.2. Applications.- 5.3. Domain Dependence.- 5.4. A Counterexample.- 5.5. Convexity.- Notes to Chapter 5.- 6 Capillary Surfaces Without Gravity.- 6.1. General Remarks.- 6.2. A Necessary Condition.- 6.3. Sufficiency Conditions.- 6.4. Sufficiency Conditions II.- 6.5. A Subsidiary Extremal Problem.- 6.6. Minimizing Sequences.- 6.7. The Limit Configuration.- 6.8. The First Variation.- 6.9. The Second Variation.- 6.10. Solution of the Jacobi Equation.- 6.11. Convex Domains.- 6.12. Continuous and Discontinuous Disappearance.- 6.13. An Example.- 6.14. Another Example.- 6.15. Remarks on the Extremals.- 6.16. Example 1.- 6.17. Example 2.- 6.18. Example 3.- 6.19. The Trapezoid.- 6.20. Tail Domains A Counterexample.- 6.21. Convexity.- 6.22. A Counterexample.- 6.23. Transition to Zero Gravity.- Notes to Chapter 6.- 7 Existence Theorems.- 7.1. Choice of Venue.- 7.2. Variational Solutions.- 7.3. Generalized Solutions.- 7.4. Construction of a Generalized Solution.- 7.5. Proof of Boundedness.- 7.6. Uniqueness.- 7.7. The Variational Condition Limiting Case.- 7.8. A Necessary and Sufficient Condition.- 7.9. A Limiting Configuration.- 7.10. The Case > 0>1.- 7.11. Application: A General Gradient Bound.- Notes to Chapter 7.- 8 The Capillary Contact Angle.- 8.1. Everyday Experience.- 8.2. The Hypothesis.- 8.3. The Horizontal Plane Preliminary Remarks.- 8.4. Necessity for ?.- 8.5. Proof that ? is Monotone.- 8.6. Geometrically Imposed Stability Bounds.- 8.7. A Further Kind of Instability.- 8.8. The Inclined Plane Preliminary Remarks.- 8.9. Integral Relations, and Impossibility of Constant Contact Angle.- 8.10. The Zero-Gravity Solution.- 8.11. Postulated Form for ?.- 8.12. Formal Analytical Solution.- 8.13. The Expansion Leading Terms.- 8.14. Computer Calculations.- 8.15. Discussion.- 8.16. Further Discussion.- Notes to Chapter 8.- 9 Identities and Isoperimetric Relations.

Abstract: I. Dirichlet's Principle and the Boundary Value Problem of Potential Theory.- 1. Dirichlet's Principle.- Definitions.- Original statement of Dirichlet's Principle.- General objection: A variational problem need not he solvable.- Minimizing sequences.- Explicit expression for Dirichlet's integral over a circle. Specific objection to Dirichlet's Principle.- Correct formulation of Dirichlet's Principle.- 2. Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk.- 3. Dirichlet's integral and quadratic functionals.- 4. Further preparation.- Convergence of a sequence of harmonic functions.- Oscillation of functions appraised by Dirichlet's integral.- Invariance of Dirichlet's integral under conformal mapping. Applications.- Dirichlet's Principle for a circle with partly free boundary.- 5. Proof of Dirichlet's Principle for general domains.- Direct methods in the calculus of variations.- Construction of the harmonic function u by a "smoothing process".- Proof that D[ul = d.- Proof that u attains prescribed boundary values.- Generalizations.- 6. Alternative proof of Dirichlet's Principle.- Fundamental integral inequality.- Solution of variational problem I.- 7. Conformal mapping of simply and doubly connected domains.- 8. Dirichlet's Principle for free boundary values. Natural boundary conditions.- II. Conformal Mapping on Parallel-Slit Domains.- 1. Introduction.- Classes of normal domains. Parallel-slit domains.- Variational problem: Motivation and formulation.- 2. Solution of variational problem II.- Construction of the function u.- Continuous dependence of the solution on the domain.- 3. Conformal mapping of plane domains on slit domains.- Mapping of k-fold connected domains.- Mapping on slit domains for domains G of infinite connectivity.- Half-plane slit domains. Moduli.- Boundary mapping.- 4. Riemann domains.- The "sewing theorem".- 5. General Riemann domains. Uniformisation.- 6. Riemann domains defined by non-overlapping cells.- 7. Conformal mapping of domains not of genus zero.- Description of slit domains not of genus zero.- The mapping theorem.- Remarks. Half-plane slit domains.- III. Plateau's Problem.- 1. Introduction.- 2. Formulation and solution of basic variational problems.- Notations.- Fundamental lemma. Solution of minimum problem.- Remarks. Semicontinuity.- 3. Proof by conformal mapping that solution is a minimal surface.- 4. First variation of Dirichlet's integral.- Variation in general space of admissible functions.- First variation in space of harmonic vectors.- Proof that stationary vectors represent minimal surfaces.- 5. Additional remarks.- Biunique correspondence of boundary points.- Relative minima.- Proof that solution of variational problem solves problem of least area.- Role of conformal mapping in solution of Plateau's problem.- 6. Unsolved problems.- Analytic extension of minimal surfaces.- Uniqueness. Boundaries spanning infinitely many minimal surfaces.- Branch points of minimal surfaces.- 7. First variation and method of descent.- 8. Dependence of area on boundary.- Continuity theorem for absolute minima.- Lengths of images of concentric circles.- Isoperimetric inequality for minimal surfaces.- Continuous variation of area of minimal surfaces.- Continuous variation of area of harmonic surfaces.- IV. The General Problem of Douglas.- 1. Introduction.- 2. Solution of variational problem for k-fold connected domains.- Formulation of problem.- Condition of cohesion.- Solution of variational problem for k-fold connected domains G and parameter domains bounded by circles.- Solution of variational problem for other classes of normal domains.- 3. Further discussion of solution.- Douglas' sufficient condition.- Lemma 4 1 and proof of theorem 4.2.- Lemma 4.2 and proof of theorem 4.1.- Remarks and examples.- 4. Generalization to higher topological structure.- Existence of solution.- Proof for topological type of Moebius strip.- Other types of parameter domains.- Identification of solutions as minimal surfaces. Properties of solution.- V. Conformal Mapping of Multiply Connected Domains.- 1. Introduction.- Objective.- First variation.- 2. Conformal mapping on circular domains.- Statement of theorem.- Statement and discussion of variational conditions.- Proof of variational conditions.- Proof that ?(w) = 0.- 3. Mapping theorems for a general class of normal domains.- Formulation of theorem.- Variational conditions.- Proof that ?(w) = 0.- 4. Conformal mapping on Riemann surfaces bounded by unit circles.- Formulation of theorem.- Variational conditions. Variation of branchpoints.- Proof that ?(w) = 0.- 5. Uniqueness theorems.- Method of uniqueness proof.- Uniqueness for Riemann surfaces with branch points.- Uniqueness for classes ? of plane domains.- Uniqueness for other classes of domains.- 6. Supplementary remarks.- First continuity theorem in conformal mapping.- Second continuity theorem. Extension of previous mapping theorems.- Further observations on conformal mapping.- 7. Existence of solution for variational problem in two dimensions.- Proof using conformal mapping of doubly connected domains.- Alternative proof. Supplementary remarks.- VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces.- 1. Introduction.- Free boundary problems.- Unstable minimal surfaces.- 2. Free boundaries. Preparations.- General remarks.- A theorem on boundary values.- 3. Minimal surfaces with partly free boundaries.- Only one arc fixed.- Remarks on Schwarz' chains.- Doubly connected minimal surfaces with one free boundary.- Multiply connected minimal surfaces with free boundaries.- 4. Minimal surfaces spanning closed manifolds.- Existence proof.- 5. Properties of the free boundary. Transversality.- Plane boundary surface. Reflection.- Surface of least area whose free boundary is not a continuous curve.- Transversality.- 6. Unstable minimal surfaces with prescribed polygonal boundaries.- Unstable stationary points for functions of N variables.- A modified variational problem.- Proof that stationary values of d(U) are stationary values for D[x].- Generalization.- Remarks on a variant of the problem and on second variation.- 7. Unstable minimal surfaces in rectifiable contours.- Preparations. Main theorem.- Remarks and generalizations.- 8. Continuity of Dirichlet's integral under transformation of x-space.- Bibliography, Chapters I to VI.- 1. Green's function and boundary value problems.- Canonical conformal mappings.- Boundary value problems of second type and Neumann's function.- 2. Dirichlet integrals for harmonic functions.- Formal remarks..- Inequalities..- Conformal transformations.- An application to the theory of univalent functions.- Discontinuities of the kernels.- An eigenvalue problem.- Comparison theory.- An extremum problem in conformal mapping.- Mapping onto a circular domain.- Orthornormal systems.- 3. Variation of the Green's function.- Hadamard's variation formula.- Interior variations.- Application to the coefficient problem for univalent functions.- Boundary variations.- Lavrentieff's method.- Method of extremal length.- Concluding remarks.- Bibliography to Appendix.- Supplementary Notes (1977).