Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces

It has always been a temptation for mathematicians to present the crystallized product of their thoughts as a deductive general theory and to relegate the individual mathematical phenomenon into the role of an example. The reader who submits to the dogmatic form will be easily indoctrinated. Enlightenment, however, must come from an understanding of motives; live mathematical development springs from specific natural problems which can be easily understood, but whose solutions are difficult and demand new methods of more general significance. The present book deals with subjects of this category. It is written in a style which, as the author hopes, expresses adequately the balance and tension between the individuality of mathematical objects and the generality of mathematical methods. The author has been interested in Dirichlet's Principle and its various applications since his days as a student under David Hilbert. Plans for writing a book on these topics were revived when Jesse Douglas' work suggested to him a close connection between Dirichlet's Principle and basic problems concerning minimal sur­ faces. But war work and other duties intervened; even now, after much delay, the book appears in a much less polished and complete form than the author would have liked.

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).