Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces

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ISBN: 9780486445526 | 0486445526
Cover: Paperback
Copyright: 11/3/2005

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Mathematical development, the author of this text observes, comes about through specific, easily understood problems that require difficult solutions and demand the use of new methods. Richard Courant employs this instructive approach in a text that balances the individuality of mathematical objects with the generality of mathematical methods.

Preface

vii

Introduction

(4)

I. Dirichlet's Principle and the Boundary Value Problem of Potential Theory

(40)

1. Dirichlet's Principle

(6)

Definitions

(1)

Original statement of Dirichlet's Principle

(1)

General objection: A variational problem need not be solvable

(2)

Minimizing sequences

(1)

Explicit expression for Dirichlet's integral over a circle. Specific objection to Dirichlet's Principle

(1)

Correct formulation of Dirichlet's Principle

(1)

2. Semicontinuity of Dirichlet's integral. Dirichlet's Principle for circular disk

(2)

3. Dirichlet's integral and quadratic functionals

(3)

4. Further preparation

(7)

Convergence of a sequence of harmonic functions

(2)

Oscillation of functions appraised by Dirichlet's integral

(2)

Invariance of Dirichlet's integral under conformal mapping Applications

(1)

Dirichlet's Principle for a circle with partly free boundary

(2)

5. Proof of Dirichlet's Principle for general domains

(8)

Direct methods in the calculus of variations

(1)

Construction of the harmonic function u by a "smoothing process"

(4)

Proof that D[u] d

(1)

Proof that u attains prescribed boundary values

(2)

Generalizations

(1)

6. Alternative proof of Dirichlet's Principle

(7)

Fundamental integral inequality

(1)

Solution of variational problem I

(6)

7. Conformal mapping of simply and doubly connected domains

(2)

8. Dirichlet's Principle for free boundary values. Natural boundary conditions

(5)

II. Conformal Mapping on Parallel-Slit Domains

(50)

1. Introduction

(6)

Classes of normal domains. Parallel-slit domains

(3)

Variational problem: Motivation and formulation

(3)

2. Solution of variational problem II

(4)

Construction of the function u

(3)

Continuous dependence of the solution on the domain

(1)

3. Conformal mapping of plane domains on slit domains

(9)

Mapping of k-fold connected domains

(2)

Mapping on slit domains for domains G of infinite connectivity

(3)

Half-plane slit domains Moduli

(1)

Boundary mapping

(2)

4. Riemann domains

(11)

Introduction

(5)

The "sewing theorem"

(6)

5. General Riemann domains. Uniformization

(3)

6. Riemann domains defined by non-overlapping cells

(2)

7. Conformal mapping of domains not of genus zero

(15)

Introduction

(1)

Description of slit domains not of genus zero

(5)

The mapping theorem

(7)

Remarks. Half-plane slit domains

(3)

III. Plateau's Problem

(46)

1. Introduction

(6)

2. Formulation and solution of basic variational problems

101

(4)

Notations

101

(1)

Fundamental lemma. Solution of minimum problem

101

(3)

Remarks. Semicontinuity

104

(1)

3. Proof by conformal mapping that solution is a minimal surface

105

(2)

4. First variation of Dirichlet's integral

107

(8)

Variation in general space of admissible functions.

107

(3)

First variation in space of harmonic vectors

110

(2)

Proof that stationary vectors represent minimal surfaces

112

(3)

5. Additional remarks

115

(3)

Biunique correspondence of boundary points

115

(1)

Relative minima

115

(1)

Proof that solution of variational problem solves problem of least area

116

(1)

Role of conformal mapping in solution of Plateau's problem

117

(1)

6. Unsolved problems

118

(5)

Analytic extension of minimal surfaces

118

(1)

Uniqueness. Boundaries spanning infinitely many minimal surfaces

119

(3)

Branch points of minimal surfaces

122

(1)

7. First variation and method of descent

123

(3)

8. Dependence of area on boundary

126

(15)

Continuity theorem for absolute minima

126

(1)

Lengths of images of concentric circles

127

(2)

Isoperimetric inequality for minimal surfaces

129

(2)

Continuous variation of area of minimal surfaces

131

(3)

Continuous variation of area of harmonic surfaces

134

(7)

IV. The General Problem of Douglas

141

(26)

1. Introduction

141

(3)

2. Solution of variational problem for k-fold connected domains

144

(5)

Formulation of problem

144

(1)

Condition of cohesion

145

(1)

Solution of variational problem for k-fold connected domains G and parameter domains bounded by circles

146

(3)

Solution of variational problem for other classes of normal domains

149

(1)

3. Further discussion of solution

149

(11)

Douglas' sufficient condition

149

(2)

Lemma 4.1 and proof of theorem 4.2

151

(2)

Lemma 4.2 and proof of theorem 4.1

153

(5)

Remarks and examples

158

(2)

4. Generalization to higher topological structure.

160

(7)

Existence of solution

160

(1)

Proof for topological type of Moebius strip

161

(3)

Other types of parameter domains

164

(1)

Identification of solutions as minimal surfaces. Properties of solution

165

(2)

V. Conformal Mapping of Multiply Connected Domains

167

(32)

1. Introduction

167

(2)

Objective

167

(1)

First variation

168

(1)

2. Conformal mapping on circular domains

169

(9)

Statement of theorem

169

(1)

Statement and discussion of variational conditions

169

(2)

Proof of variational conditions

171

(4)

Proof that Φ(w) = 0

175

(3)

3. Mapping theorems for a general class of normal domains

178

(5)

Formulation of theorem

178

(1)

Variational conditions

179

(1)

Proof that Φ(w) = 0

180

(3)

4. Conformal mapping on Riemann surfaces bounded by unit circles

183

(4)

Formulation of theorem

183

(1)

Variational conditions. Variation of branchpoints

184

(2)

Proof that Φ(w) = 0

186

(1)

5. Uniqueness theorems

187

(4)

Method of uniqueness proof

187

(1)

Uniqueness for Riemann surfaces with branch points

188

(1)

Uniqueness for classes 91 of plane domains

188

(2)

Uniqueness for other classes of domains

190

(1)

6. Supplementary remarks

191

(1)

First continuity theorem in conformal mapping

191

(1)

Second continuity theorem. Extension of previous mapping theorems

191

(1)

Further observations on conformal mapping

192

(1)

7. Existence of solution for variational problem in two dimensions

192

(7)

Proof using conformal mapping of doubly connected domains

192

(5)

Alternative proof. Supplementary remarks

197

(2)

VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Surfaces

199

(46)

1. Introduction

199

(2)

Free boundary problems

199

(1)

Unstable minimal surfaces

200

(1)

2. Free boundaries. Preparations

201

(5)

General remarks

201

(1)

A theorem on boundary values

202

(4)

3. Minimal surfaces with partly free boundaries

206

(7)

Only one arc fixed

206

(2)

Remarks on Schwarz' chains

208

(1)

Doubly connected minimal surfaces with one free boundary

209

(2)

Multiply connected minimal surfaces with free boundaries

211

(2)

4. Minimal surfaces spanning closed manifolds

213

(5)

Introduction

213

(1)

Existence proof

214

(4)

5. Properties of the free boundary. Transversality

218

(5)

Plane boundary surface. Reflection

218

(2)

Surface of least area whose free boundary is not a continuous curve

220

(2)

Transversality

222

(1)

6. Unstable minimal surfaces with prescribed polygonal boundaries

223

(13)

Unstable stationary points for functions of N variables

223

(3)

A modified variational problem

226

(6)

Proof that stationary values of d(U) are stationary values for D[ ]

232

(1)

Generalization

233

(2)

Remarks on a variant of the problem and on second variation

235

(1)

7. Unstable minimal surfaces in rectifiable contours

236

(5)

Preparations. Main theorem

236

(4)

Remarks and generalizations

240

(1)

8. Continuity of Dirichlet's integral under transformation of r-space.

241

(4)

Bibliography, Chapters Ito VI

245

(4)

Appendix. Some Recent Developments in the Theory of Conformal Mapping. By M. SCHIFFER

249

(68)

1. Green's function and boundary value problems

249

(17)

Canonical conformal mappings

253

(6)

Boundary value problems of second type and Neumann's function

259

(7)

2. Dirichlet integrals for harmonic functions

266

(26)

Formal remarks

266

(2)

The kernels K and L

268

(5)

Inequalities

273

(2)

Conformal transformations

275

(1)

An application to the theory of univalent functions

276

(1)

Discontinuities of the kernels

277

(1)

An eigenvalue problem

278

(3)

Kernel functions for the class o

281

(2)

Comparison theory

283

(6)

An extremum problem in conformal mapping

289

(1)

Mapping onto a circular domain

290

(1)

Orthornormal systems

291

(1)

3. Variation of the Green's function

292

(25)

Hadamard's variation formula

292

(6)

Interior variations

298

(2)

Application to the coefficient problem for univalent functions

300

(6)

Boundary variations

306

(4)

Lavrentieff's method

310

(3)

Method of extremal length

313

(4)

Concluding remarks

317

(2)

Bibliography to Appendix

319

(6)

Index

325

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Dirichlet's Principle, Conformal Mapping, and Minimal Surfaces

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