| Preface |
| Introduction |
| Dirichlet's Principle and the Boundary Value Problem of Potential Theory / I： |
| Dirichlet's Principle / 1： |
| Definitions |
| Original statement of Dirichlet's Principle |
| General objection: A variational problem need not be solvable |
| Minimizing sequences |
| Explicit expression for Dirichlet's integral over a circle. Specific objection to Dirichlet's Principle |
| Correct formulation of Dirichlet's Principle |
| Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk / 2： |
| Dirichlet's integral and quadratic functionals / 3： |
| Further preparation / 4： |
| 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 |
| Proof of Dirichlet's Principle for general domains / 5： |
| Direct methods in the calculus of variations |
| Construction of the harmonic function u by a "smoothing process" |
| Proof that D[u] = d |
| Proof that u attains prescribed boundary values |
| Generalizations |
| Alternative proof of Dirichlet's Principle / 6： |
| Fundamental integral inequality |
| Solution of variational problem I |
| Conformal mapping of simply and doubly connected domains / 7： |
| Dirichlet's Principle for free boundary values. Natural boundary conditions / 8： |
| Conformal Mapping on Parallel-Slit Domains / II： |
| Classes of normal domains. Parallel-slit domains |
| Variational problem: Motivation and formulation |
| Solution of variational problem II |
| Construction of the function u |
| Continuous dependence of the solution on the domain |
| 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 |
| Riemann domains |
| The "sewing theorem" |
| General Riemann domains. Uniformization |
| Riemann domains defined by non-overlapping cells |
| Conformal mapping of domains not of genus zero |
| Description of slit domains not of genus zero |
| The mapping theorem |
| Remarks. Half-plane slit domains |
| Plateau's Problem / III： |
| Formulation and solution of basic variational problems |
| Notations |
| Fundamental lemma. Solution of minimum problem |
| Remarks. Semicontinuity |
| Proof by conformal mapping that solution is a minimal surface |
| 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 |
| 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 |
| Unsolved problems |
| Analytic extension of minimal surfaces |
| Uniqueness. Boundaries spanning infinitely many minimal surfaces |
| Branch points of minimal surfaces |
| First variation and method of descent |
| 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 |
| The General Problem of Douglas / IV： |
| 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 |
| 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 |
| 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 |
| Conformal Mapping of Multiply Connected Domains / V： |
| Objective |
| First variation |
| Conformal mapping on circular domains |
| Statement of theorem |
| Statement and discussion of variational conditions |
| Proof of variational conditions |
| Proof that [phi](w) = 0 |
| Mapping theorems for a general class of normal domains |
| Formulation of theorem |
| Variational conditions |
| Conformal mapping on Riemann surfaces bounded by unit circles |
| Variational conditions. Variation of branchpoints |
| Uniqueness theorems |
| Method of uniqueness proof |
| Uniqueness for Riemann surfaces with branch points |
| Uniqueness for classes [characters not reproducible] of plane domains |
| Uniqueness for other classes of domains |
| Supplementary remarks |
| First continuity theorem in conformal mapping |
| Second continuity theorem. Extension of previous mapping theorems |
| Further observations on conformal mapping |
| Existence of solution for variational problem in two dimensions |
| Proof using conformal mapping of doubly connected domains |
| Alternative proof. Supplementary remarks |
| Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces / VI： |
| Free boundary problems |
| Unstable minimal surfaces |
| Free boundaries. Preparations |
| General remarks |
| A theorem on boundary values |
| 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 |
| Minimal surfaces spanning closed manifolds |
| Existence proof |
| Properties of the free boundary. Transversality |
| Plane boundary surface. Reflection |
| Surface of least area whose free boundary is not a continuous curve |
| Transversality |
| 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[characters not reproducible] |
| Generalization |
| Remarks on a variant of the problem and on second variation |
| Unstable minimal surfaces in rectifiable contours |
| Preparations. Main theorem |
| Remarks and generalizations |
| Continuity of Dirichlet's integral under transformation of [characters not reproducible]-space |
| Bibliography, Chapters I to VI |
| Some Recent Developments in the Theory of Conformal Mapping / M. SchifferAppendix： |
| Green's function and boundary value problems |
| Canonical conformal mappings |
| Boundary value problems of second type and Neumann's function |
| Dirichlet integrals for harmonic functions |
| Formal remarks |
| The kernels K and L |
| Inequalities |
| Conformal transformations |
| An application to the theory of univalent functions |
| Discontinuities of the kernels |
| An eigenvalue problem |
| Kernel functions for the class [characters not reproducible] |
| Comparison theory |
| An extremum problem in conformal mapping |
| Mapping onto a circular domain |
| Orthornormal systems |
| 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 |
| Index |
| Preface |
| Introduction |
| Dirichlet's Principle and the Boundary Value Problem of Potential Theory / I： |
図書
