Space mappings with bounded distortion:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, R.I.
American Math. Society
1989
|
| Series: | Translations of mathematical monographs
73 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Aus d. Russ. übers. |
| Physical Description: | XV, 362 S. |
| ISBN: | 0821845268 |
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MARC
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| 100 | 1 | |a Rešetnjak, Jurij G. |e Verfasser |4 aut | |
| 240 | 1 | 0 | |a Prostranstvennye otobraženija s ograničennym iskaženiem |
| 245 | 1 | 0 | |a Space mappings with bounded distortion |c Yu. G. Reshetnyak |
| 264 | 1 | |a Providence, R.I. |b American Math. Society |c 1989 | |
| 300 | |a XV, 362 S. | ||
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Record in the Search Index
| _version_ | 1804117110465495040 |
|---|---|
| adam_text | Contents
Foreword to the English translation xi
From the Author xiii
CHAPTER I. Introduction 1
§ 1. Some facts from the theory of functions of a real variable 1
1.1. Sets in R 1
1.2. Classes of functions in R 4
1.3. Differentiation of measures on the space R . Lebesgue
points of a function and points of density of a subset of
R 6
1.4. Approximation of integrable functions by smooth
functions 10
§2. Functions with generalized derivatives 12
2.1. Definition of a function with generalized derivatives 12
2.2. Sobolev imbedding theorems 15
2.3. Tests for a function to belong to the class WlpXoc(U) 17
2.4. Transformations of functions with generalized
derivatives 21
2.5. Dependence of the coefficients in the imbedding
theorems on the size of the domain 24
2.6. A theorem on differentiability of Wx loc functions
almost everywhere 25
§3. Mobius transformations 28
3.1. Motions and similarity transformations of a Euclidean
space 28
3.2. Mobius transformations. Definitions 33
3.3. Mobius transformations and cross ratios. Construction
of Mobius transformations 36
3.4. Mobius transformations and spheres 39
V
vi CONTENTS
3.5. The hypersphere bundle and linear representations of
Mobius transformations 44
§4. Definition of a mapping with bounded distortion 53
4.1. Orthogonal invariants of linear mappings of Euclidean
spaces. A measure of nonorthogonality for a linear
mapping 53
4.2. Mappings with bounded distortion 61
4.3. Examples of mappings with bounded distortion 63
§5. Mappings with bounded distortion on Riemannian spaces 67
5.1. Riemannian metrics in domains in R 67
5.2. Mappings with bounded distortion on Riemannian
spaces 72
CHAPTER II. Main facts in the theory of mappings with bounded
distortion 79
§ 1. Estimates of the moduli of continuity and differentiability
almost everywhere of mappings with bounded distortion 79
1.1. Some auxiliary facts 79
1.2. An estimate of the modulus of continuity of a mapping
with bounded distortion 82
1.3. Differentiability almost everywhere of mappings with
bounded distortion 83
§2. Some facts about continuous mappings on R 85
2.1. The degree of a mapping 85
2.2. The degree of a mapping and exterior differential forms 90
2.3. Change of variables in a multiple integral 93
§3. Conformal capacity 103
3.1. The capacity of a capacitor 103
3.2. Sets of zero capacity 110
3.3. The concept of a Hausdorff measure. Cartan s lemma 114
3.4. Capacity and Hausdorff measures 118
3.5. Estimates of the capacity of certain capacitors 120
§4. The concept of the generalized differential of an exterior
form 129
4.1. General facts about exterior forms 129
4.2. The concept of generalized differential of an exterior
form 131
4.3. Properties of the generalized differential of an exterior
form 133
CONTENTS vii
4.4. The homomorphism induced on the algebra of exterior
forms by a mapping of the domain 134
4.5. Weak convergence of sequences of exterior forms 138
§5. Mappings with bounded distortion and elliptic differential
equations 141
5.1. A description of a certain class of functionals of the
calculus of variations 141
5.2. Variational properties of mappings with bounded
distortion 144
5.3. The classes W*(U/A) and W],(U/A) 149
5.4. The Dirichlet problem, extremal functions, and
generalized solutions of the Euler equation for
functionals of the calculus of variations 158
5.5. The maximum principle for extremals of functionals of
the calculus of variations 161
5.6. Harnack s inequality and its corollaries 163
5.7. The concept of the flow of a stationary function in a
capacitor 164
5.8. The set of singular points of stationary functions for
functionals of the calculus of variations 168
5.9. Liouville s theorem on conformal mappings in space 171
5.10. The property of quasi invariance of conformal capacity 172
§6. Topological properties of mappings with bounded distortion 173
6.1. Continuous mappings with nonnegative Jacobian 173
6.2. Satisfaction of condition N for mappings with bounded
distortion 176
6.3. Topological properties of mappings with bounded
distortion 182
6.4. A theorem on removable singularities 187
6.5. On the method of moduli 188
6.6. Bi Lipschitz mappings 190
§7. Local structure of mappings with bounded distortion 194
7.1. Preliminary remarks 194
7.2. Some estimates of a solution of an elliptic equation
having one singular point 196
7.3. A measure of the distortion of a small sphere under a
mapping with bounded distortion 200
7.4. Behavior of a mapping with bounded distortion near an
arbitrary point of the domain 201
viii CONTENTS
§8. Characterization of mappings with bounded distortion by
the property of quasiconformality 204
8.1. The concept of a mapping which is quasiconformal at a
point and in a domain 204
8.2. Differentiability almost everywhere of quasiconformal
T mappings 205
8.3. The condition of absolute continuity for a real function
of a single variable 208
8.4. The analytic nature of quasiconformal T mappings 210
8.5. Main result 213
8.6. Homeomorphic quasiconformal mappings 215
§9. Sequences of mappings with bounded distortion 216
9.1. A theorem on local boundedness of sequences of
mappings with bounded distortion 216
9.2. A theorem on the limit of a sequence of mappings with
bounded distortion 218
9.3. A sufficient condition for relative compactness of a
family of mappings with bounded distortion 220
§10. The set of branch points of a mapping with bounded
distortion and locally homeomorphic mappings 221
10.1. The measure of the set of branch points 221
10.2. Some lemmas on local homeomorphisms 224
10.3. The measure of the image of the set of branch points
for mappings with bounded distortion 227
10.4. A local homeomorphism theorem 229
10.5. A theorem on the radius of injectivity 234
§11. Extremal properties of mappings with bounded distortion 240
11.1. The homomorphism generated on the algebra of
exterior forms by a mapping with bounded distortion 240
11.2. Main theorem 242
§12. Some further results 246
12.1. Classes of domains in R 246
12.2. Stability in the Liouville theorem on conformal
mappings of a space and related questions 251
12.3. Stability of isometric and Lorentz transformations 260
12.4. Quasiconformal and quasi isometric deformations.
Semigroups of quasiconformal transformations 266
12.5. Mappings with distortion coefficient close to 1 277
CONTENTS ix
12.6. The general concept of stability classes 283
12.7. A characterization of quasiconformal mappings as
mappings preserving the space W 287
CHAPTER III. Some results in the theory of functions of a real
variable and the theory of partial differential
equations 289
§1. Functions with bounded mean oscillation 289
§2. Harnack s inequality for quasilinear elliptic equations 295
2.1. Preliminary remarks 295
2.2. Main inequalities 296
2.3. Consequences of the integral inequalities in §2.2 300
2.4. Boundedness of generalized solutions of equation (2.3).
Harnack s inequality 304
§3. Theorems on semicontinuity and convergence with a
functional for functional of the calculus of variations 310
3.1. Weak convergence of sequences of functions in measure
spaces 310
3.2. Some lemmas about convex functions 312
3.3. Theorems about semicontinuity of functional of the
calculus of variations 315
3.4. Corollaries to Theorems 3.1 and 3.2 319
3.5. The convex envelope of a function 321
3.6. A theorem on convergence with a functional 325
3.7. Corollaries to the theorem on convergence with a
functional 329
§4. Some properties of functions with generalized derivatives 330
4.1. A theorem on differentiability almost everywhere 330
4.2. Proof of Lemma 1.1 in Chapter II 335
4.3. An estimate of the modulus of continuity of a
monotone function of class Wj, 338
§5. On the degree of a mapping 341
Bibliography 347
|
| any_adam_object | 1 |
| author | Rešetnjak, Jurij G. |
| author_facet | Rešetnjak, Jurij G. |
| author_role | aut |
| author_sort | Rešetnjak, Jurij G. |
| author_variant | j g r jg jgr |
| building | Verbundindex |
| bvnumber | BV002757742 |
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| ctrlnum | (OCoLC)246829122 (DE-599)BVBBV002757742 |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV002757742 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:49:11Z |
| institution | BVB |
| isbn | 0821845268 |
| language | English |
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| physical | XV, 362 S. |
| publishDate | 1989 |
| publishDateSearch | 1989 |
| publishDateSort | 1989 |
| publisher | American Math. Society |
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| series | Translations of mathematical monographs |
| series2 | Translations of mathematical monographs |
| spelling | Rešetnjak, Jurij G. Verfasser aut Prostranstvennye otobraženija s ograničennym iskaženiem Space mappings with bounded distortion Yu. G. Reshetnyak Providence, R.I. American Math. Society 1989 XV, 362 S. txt rdacontent n rdamedia nc rdacarrier Translations of mathematical monographs 73 Aus d. Russ. übers. Konforme Abbildung (DE-588)4164968-0 gnd rswk-swf Konforme Abbildung (DE-588)4164968-0 s DE-604 Translations of mathematical monographs 73 (DE-604)BV000002394 73 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001762878&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rešetnjak, Jurij G. Space mappings with bounded distortion Translations of mathematical monographs Konforme Abbildung (DE-588)4164968-0 gnd |
| subject_GND | (DE-588)4164968-0 |
| title | Space mappings with bounded distortion |
| title_alt | Prostranstvennye otobraženija s ograničennym iskaženiem |
| title_auth | Space mappings with bounded distortion |
| title_exact_search | Space mappings with bounded distortion |
| title_full | Space mappings with bounded distortion Yu. G. Reshetnyak |
| title_fullStr | Space mappings with bounded distortion Yu. G. Reshetnyak |
| title_full_unstemmed | Space mappings with bounded distortion Yu. G. Reshetnyak |
| title_short | Space mappings with bounded distortion |
| title_sort | space mappings with bounded distortion |
| topic | Konforme Abbildung (DE-588)4164968-0 gnd |
| topic_facet | Konforme Abbildung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001762878&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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