Dirichlet's principle, conformal mapping, and minimal surfaces:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Interscience Publishers
1967
|
| Edition: | Third printing |
| Series: | Pure and Applied Mathematics
3 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 330 Seiten |
Staff View
MARC
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| 100 | 1 | |a Courant, Richard |d 1888-1972 |e Verfasser |0 (DE-588)118522442 |4 aut | |
| 245 | 1 | 0 | |a Dirichlet's principle, conformal mapping, and minimal surfaces |c R. Courant, Institute for Mathematics and Mechanics, New York University, New York, with an appendix by M. Schiffer, Princeton University and University of Jerusalem |
| 250 | |a Third printing | ||
| 264 | 1 | |a New York, NY |b Interscience Publishers |c 1967 | |
| 300 | |a XIII, 330 Seiten | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface vii
Introduction 1
I. Dirichlet s Principle and the Boundary Value Problem of Potential
Theory 5
1. Dirichlet s Principle 5
Definitions 5
Original statement of Dirichlet s Principle 6
General objection: A variational problem need not be
solvable 6
Minimizing sequences . 8
Explicit expression for Dirichlet s integral over a circle. Spe¬
cific objection to Dirichlet s Principle 9
Correct formulation of Dirichlet s Principle 10
2. Semicontinuity of Dirichlet s integral. Dirichlet s Principle for cir¬
cular disk 11
3. Dirichlet s integral and quadratic functionals 13
4. Further preparation 16
Convergence of a sequence of harmonic functions 16
Oscillation of functions appraised by Dirichlet s integral 18
Invariance of Dirichlet s integral under conformal mapping.
Applications 20
Dirichlet s Principle for a circle with partly free boundary 21
5. Proof of Dirichlet s Principle for general domains 23
Direct methods in the calculus of variations 23
Construction of the harmonic function u by a smoothing proc¬
ess 24
Proof that D[u] = d 28
Proof that u attains prescribed boundary values 28
Generalizations 30
6. Alternative proof of Dirichlet s Principle 31
Fundamental integral inequality 31
Solution of variational problem 1 32
7. Conformal mapping of simply and doubly connected domains... 38
8. Dirichlet s Principle for free boundary values. Natural boundary
conditions 40
ix
X CONTENTS
II. Con formal Mapping on Parallel-Slit Domains 45
1. Introduction 45
Classes of normal domains. Parallel-slit domains 45
Variational problem: Motivation and formulation 48
2. Solution of variational problem II 51
Construction of the function u 51
Continuous dependence of the solution on the domain 54
3. Conformal mapping of plane domains on slit domains 55
Mapping of k-fold connected domains 56
Mapping on slit domains for domains G of infinite con¬
nectivity 58
Half-plane slit domains. Moduli 61
Boundary mapping 62
4. Riemann domains 64
Introduction 64
The sewing theorem 69
5. General Riemann domains. Uniformization 75
6. Riemann domains defined by non-overlapping cells 78
7. Conformal mapping of domains not of genus zero 80
Introduction 80
Description of slit domains not of genus zero 80
The mapping theorem 85
Remarks. Half-plane slit domains 92
III. Plateau s Problem 95
1. Introduction 95
2. Formulation and solution of basic variational problems 101
Notations 101
Fundamental lemma. Solution of minimum problem 101
Remarks. Semicontinuity 104
3. Proof by conformal mapping that solution is a minimal surface 105
4. First variation of Dirichlet s integral 107
Variation in general space of admissible functions 107
First variation in space of harmonic vectors 110
Proof that stationary vectors represent minimal surfaces 112
5. Additional remarks 115
Biunique correspondence of boundary points 115
Relative minima 115
Proof that solution of variational problem solves problem of
least area 116
Role of conformal mapping in solution of Plateau s problem.... 117
6. Unsolved problems 118
Analytic extension of minimal surfaces 118
Uniqueness. Boundaries spanning infinitely many minimal
surfaces 119
Branch points of minimal surfaces 122
CONTENTS xi
III. Plateau s Problem—Continued
7. First variation and method of descent 123
8. Dependence of area on boundary 126
Continuity theorem for absolute minima 126
Lengths of images of concentric circles 127
Isoperimetric inequality for minimal surfaces 129
Continuous variation of area of minimal surfaces 131
Continuous variation of area of harmonic surfaces 134
IV. The General Problem of Douglas 141
1. Introduction 141
2. Solution of variational problem for k-fold connected domains.... 144
Formulation of problem 144
Condition of cohesion 145
Solution of variational problem for k-fold connected domains
G and parameter domains bounded by circles 146
Solution of variational problem for other classes of normal do¬
mains 149
3. Further discussion of solution 149
Douglas sufficient condition 149
Lemma 4.1 and proof of theorem 4.2 151
Lemma 4.2 and proof of theorem 4.1 153
Remarks and examples 158
4. Generalization to higher topological structure 160
Existence of solution 160
Proof for topological type of Moebius strip 161
Other types of parameter domains 164
Identification of solutions as minimal surfaces. Properties of
solution 165
V. Conformal Mapping of Multiply Connected Domains 167
1. Introduction 167
Objective 167
First variation 168
2. Conformal mapping on circular domains 169
Statement of theorem 169
Statement and discussion of variational conditions 169
Proof of variational conditions 171
Proof that *(u ) = 0 175
3. Mapping theorems for a general class of normal domains 178
Formulation of theorem 178
Variational conditions 179
Proof that *(u ) - 0 180
4. Conformal mapping on Riemann surf aces bounded by unit circles... 183
Formulation of theorem 183
xij CONTENTS
V. Conformal Mapping of Multiply Connected Domains—Continued
Variational conditions. Variation of branchpoints 184
Proof that #(u ) =0 186
5. Uniqueness theorems 187
Method of uniqueness proof 187
Uniqueness for Riemann surfaces with branch points 188
Uniqueness for classes 9t of plane domains 188
Uniqueness for other classes of domains 190
6. Supplementary remarks 191
First continuity theorem in conformal mapping 191
Second continuity theorem. Extension of previous mapping
theorems 191
Further observations on eonformal mapping 192
7. Existence of solution for variational problem in two dimensions.... 192
Proof using conformal mapping of doubly connected domains .. 192
Alternative proof. Supplementary remarks 197
VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Sur¬
faces 199
1. Introduction 199
Free boundary problems 199
Unstable minimal surfaces 200
2. Free boundaries. Preparations 201
General remarks 201
A theorem on boundary values 202
3. Minimal surfaces with partly free boundaries 206
Only one arc fixed 206
Remarks on Schwarz chains 208
Doubly connected minimal surfaces with one free boundary.... 209
Multiply connected minimal surfaces with free boundaries 211
4. Minimal surfaces spanning closed manifolds 213
Introduction 213
Existence proof 214
5. Properties of the free boundary. Transversality 218
Plane boundary surface. Reflection 218
Surface of least area whose free boundary is not a continuous
curve 220
Transversality 222
6. Unstable minimal surfaces with prescribed polygonal boundaries... 223
Unstable stationary points for functions of N variables 223
A modified variational problem 226
Proof that stationary values of d(U) are stationary values for
£ [r] 232
Generalization 233
Remarks on a variant of the problem and on second variation... 235
7. Unstable minimal surfaces in rectifiable contours 236
CONTENTS Xlii
VI. Minimal Surfaces with Free Boundaries and Unstable Minimal Sur¬
faces—Continued
Preparations. Main theorem 236
Remarks and generalizations 240
8. Continuity of Diriehlet s integral under transformation of j-space. 241
Bibliography, Chapters I to VI 245
Appendix. Some Recent Developments in the Theory of Conformal
Mapping. By M. Schiffer 249
1. Green s function and boundary value problems 249
Canonical conformal mappings 253
Boundary value problems of second type and Neu¬
mann s function 259
2. Dirichlet integrals for harmonic functions 266
Formal remarks 266
The kernels K and L 268
Inequalities 273
Conformal transformations 275
An application to the theory of univalent functions. 276
Discontinuities of the kernels 277
An eigenvalue problem 278
Kernel functions for the class go 281
Comparison theory 283
An extremum problem in conformal mapping 289
Mapping onto a circular domain 290
Orthornormal systems 291
3. Variation of the Green s function 292
Hadamard s variation formula 292
Interior variations 298
Application to the coefficient problem for univalent
functions 300
Boundary variations 306
Lavrentieff s method 310
Method of extremal length 313
Concluding remarks 317
Bibliography to Appendix 319
Index 325
|
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| author | Courant, Richard 1888-1972 |
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| edition | Third printing |
| format | Book |
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| indexdate | 2024-07-09T16:56:15Z |
| institution | BVB |
| language | English |
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| physical | XIII, 330 Seiten |
| publishDate | 1967 |
| publishDateSearch | 1967 |
| publishDateSort | 1967 |
| publisher | Interscience Publishers |
| record_format | marc |
| series | Pure and Applied Mathematics |
| series2 | Pure and Applied Mathematics |
| spelling | Courant, Richard 1888-1972 Verfasser (DE-588)118522442 aut Dirichlet's principle, conformal mapping, and minimal surfaces R. Courant, Institute for Mathematics and Mechanics, New York University, New York, with an appendix by M. Schiffer, Princeton University and University of Jerusalem Third printing New York, NY Interscience Publishers 1967 XIII, 330 Seiten txt rdacontent n rdamedia nc rdacarrier Pure and Applied Mathematics 3 Dirichletsches Prinzip (DE-588)4150142-1 gnd rswk-swf Dirichlet-Problem (DE-588)4129762-3 gnd rswk-swf Konforme Abbildung (DE-588)4164968-0 gnd rswk-swf Minimalfläche (DE-588)4127814-8 gnd rswk-swf Potenzialtheorie (DE-588)4046939-6 gnd rswk-swf Minimalfläche (DE-588)4127814-8 s DE-604 Konforme Abbildung (DE-588)4164968-0 s Dirichletsches Prinzip (DE-588)4150142-1 s Potenzialtheorie (DE-588)4046939-6 s Dirichlet-Problem (DE-588)4129762-3 s 1\p DE-604 Pure and Applied Mathematics 3 (DE-604)BV010179752 3 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004550270&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Courant, Richard 1888-1972 Dirichlet's principle, conformal mapping, and minimal surfaces Pure and Applied Mathematics Dirichletsches Prinzip (DE-588)4150142-1 gnd Dirichlet-Problem (DE-588)4129762-3 gnd Konforme Abbildung (DE-588)4164968-0 gnd Minimalfläche (DE-588)4127814-8 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| subject_GND | (DE-588)4150142-1 (DE-588)4129762-3 (DE-588)4164968-0 (DE-588)4127814-8 (DE-588)4046939-6 |
| title | Dirichlet's principle, conformal mapping, and minimal surfaces |
| title_auth | Dirichlet's principle, conformal mapping, and minimal surfaces |
| title_exact_search | Dirichlet's principle, conformal mapping, and minimal surfaces |
| title_full | Dirichlet's principle, conformal mapping, and minimal surfaces R. Courant, Institute for Mathematics and Mechanics, New York University, New York, with an appendix by M. Schiffer, Princeton University and University of Jerusalem |
| title_fullStr | Dirichlet's principle, conformal mapping, and minimal surfaces R. Courant, Institute for Mathematics and Mechanics, New York University, New York, with an appendix by M. Schiffer, Princeton University and University of Jerusalem |
| title_full_unstemmed | Dirichlet's principle, conformal mapping, and minimal surfaces R. Courant, Institute for Mathematics and Mechanics, New York University, New York, with an appendix by M. Schiffer, Princeton University and University of Jerusalem |
| title_short | Dirichlet's principle, conformal mapping, and minimal surfaces |
| title_sort | dirichlet s principle conformal mapping and minimal surfaces |
| topic | Dirichletsches Prinzip (DE-588)4150142-1 gnd Dirichlet-Problem (DE-588)4129762-3 gnd Konforme Abbildung (DE-588)4164968-0 gnd Minimalfläche (DE-588)4127814-8 gnd Potenzialtheorie (DE-588)4046939-6 gnd |
| topic_facet | Dirichletsches Prinzip Dirichlet-Problem Konforme Abbildung Minimalfläche Potenzialtheorie |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=004550270&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV010179752 |
| work_keys_str_mv | AT courantrichard dirichletsprincipleconformalmappingandminimalsurfaces |