Dirichlet’s Principle, Conformal Mapping, and Minimal Surfaces

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).