Elliptic partial differential equations of second order:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
1983
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
224 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XIII, 513 S. |
| ISBN: | 354013025X 038713025X |
Internformat
MARC
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| 245 | 1 | 0 | |a Elliptic partial differential equations of second order |c David Gilbarg ; Neil S. Trudinger |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 1983 | |
| 300 | |a XIII, 513 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 224 | |
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| 650 | 7 | |a Equations du second degré |2 ram | |
| 650 | 7 | |a Opérateurs elliptiques |2 ram | |
| 650 | 4 | |a Équations différentielles elliptiques | |
| 650 | 7 | |a Équations différentielles elliptiques |2 ram | |
| 650 | 7 | |a équation 2ème ordre |2 inriac | |
| 650 | 7 | |a équation différentielle elliptique |2 inriac | |
| 650 | 7 | |a équation elliptique |2 inriac | |
| 650 | 7 | |a équation linéaire |2 inriac | |
| 650 | 7 | |a équation quasilinéaire |2 inriac | |
| 650 | 4 | |a Differential equations, Elliptic | |
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Datensatz im Suchindex
| _version_ | 1804114818977759232 |
|---|---|
| adam_text | Table of Contents
Chapter 1. Introduction 1
Part I. Linear Equations 11
Chapter 2. Laplace s Equation 13
2.1. The Mean Value Inequalities 13
2.2. Maximum and Minimum Principle 15
2.3. The Harnack Inequality 16
2.4. Green s Representation 17
2.5. The Poisson Integral 19
2.6. Convergence Theorems 21
2.7. Interior Estimates of Derivatives 22
2.8. The Dirichlet Problem; the Method of Subharmonic Functions . 23
2.9. Capacity 27
Problems 28
Chapter 3. The Classical Maximum Principle 31
3.1. The Weak Maximum Principle 32
3.2. The Strong Maximum Principle 33
3.3. Apriori Bounds 36
3.4. Gradient Estimates for Poisson s Equation 37
3.5. A Harnack Inequality 41
3.6. Operators in Divergence Form 45
Notes 46
Problems 47
Chapter 4. Poisson s Equation and the Newtonian Potential 51
4.1. Holder Continuity 51
4.2. The Dirichlet Problem for Poisson s Equation 54
4.3. Holder Estimates for the Second Derivatives 56
4.4. Estimates at the Boundary 64
X Table of Contents
4.5. Holder Estimates for the First Derivatives 67
Notes 70
Problems 70
Chapter 5. Banach and Hilbert Spaces 73
5.1. The Contraction Mapping Principle 74
5.2. The Method of Continuity 74
5.3. The Fredholm Alternative 75
5.4. Dual Spaces and Adjoints 79
5.5. Hilbert Spaces 80
5.6. The Projection Theorem 81
5.7. The Riesz Representation Theorem 82
5.8. The Lax Milgram Theorem 83
5.9. The Fredholm Alternative in Hilbert Spaces 83
5.10. Weak Compactness 85
Notes 85
Problems 86
Chapter 6. Classical Solutions; the Schauder Approach 87
6.1. The Schauder Interior Estimates 89
6.2. Boundary and Global Estimates 94
6.3. The Dirichlet Problem 100
6.4. Interior and Boundary Regularity 109
6.5. An Alternative Approach 112
6.6. Non Uniformly Elliptic Equations 116
6.7. Other Boundary Conditions; the Oblique Derivative Problem . . 120
6.8. Appendix 1: Interpolation Inequalities 130
6.9. Appendix 2: Extension Lemmas 136
Notes 138
Problems 141
Chapter 7. Sobolev Spaces . 144
7.1. U Spaces 145
7.2. Regularization and Approximation by Smooth Functions ... 147
7.3. Weak Derivatives. 149
7.4. The Chain Rule 151
7.5. The Wk » Spaces 153
7.6. Density Theorems 154
7.7. Imbedding Theorems . 155
7.8. Potential Estimates and Imbedding Theorems 159
7.9. The Morrey and John Nirenberg Estimates 164
7.10. Compactness Results . 167
Table of Contents xi
7.11. Difference Quotients 168
7.12. Extension and Interpolation 169
Notes 173
Problems 173
Chapter 8. Generalized Solutions and Regularity 177
8.1. The Weak Maximum Principle 179
8.2. Solvability of the Dirichlet Problem 181
8.3. Differentiability of Weak Solutions 183
8.4. Global Regularity 186
8.5. Global Boundedness of Weak Solutions 188
8.6. Local Properties of Weak Solutions 194
8.7. The Strong Maximum Principle 198
8.8. The Harnack Inequality 199
8.9. Holder Continuity 200
8.10. Local Estimates at the Boundary 202
8.11. Holder Estimates for the First Derivatives 209
8.12. The Eigenvalue Problem 212
Notes 214
Problems 216
Chapter 9. Strong Solutions 219
9.1. Maximum Principles for Strong Solutions 220
9.2. LP Estimates: Preliminary Analysis 225
9.3. The Marcinkiewicz Interpolation Theorem 227
9.4. The Calderon Zygmund Inequality 230
9.5. L Estimates 235
9.6. The Dirichlet Problem 241
9.7. A Local Maximum Principle 244
9.8. Holder and Harnack Estimates 246
9.9. Local Estimates at the Boundary 250
Notes 252
Problems 253
Part II. Quasilinear Equations 257
Chapter 10. Maximum and Comparison Principles 259
10.1. The Comparison Principle 263
10.2. Maximum Principles 264
10.3. A Counterexample 267
10.4. Comparison Principles for Divergence Form Operators .... 268
10.5. Maximum Principles for Divergence Form Operators 271
Notes 277
Problems 277
xii Table of Contents
Chapter 11. Topological Fixed Point Theorems and Their Application . . 279
11.1. The Schauder Fixed Point Theorem 279
11.2. The Leray Schauder Theorem: a Special Case 280
11.3. An Application 282
11.4. The Leray Schauder Fixed Point Theorem 286
11.5. Variational Problems 288
Notes 293
Chapter 1 2. Equations in Two Variables 294
12.1. Quasiconformal Mappings 294
12.2. Holder Gradient Estimates for Linear Equations 300
12.3. The Dirichlet Problem for Uniformly Elliptic Equations .... 304
12.4. Non Uniformly Elliptic Equations 309
Notes 315
Problems 317
Chapter 13. Holder Estimates for the Gradient 319
13.1. Equations of Divergence Form 319
13.2. Equations in Two Variables 323
13.3. Equations of General Form; the Interior Estimate 324
13.4. Equations of General Form; the Boundary Estimate 328
13.5. Application to the Dirichlet Problem 331
Notes 332
Chapter 14. Boundary Gradient Estimates 333
14.1. General Domains 335
14.2. Convex Domains 337
14.3. Boundary Curvature Conditions 341
14.4. Non Existence Results 347
14.5. Continuity Estimates 353
14.6. Appendix: Boundary Curvatures and the Distance Function . . 354
Notes 357
Problems 358
Chapter 15. Global and Interior Gradient Bounds 359
15.1. A Maximum Principle for the Gradient 359
15.2. The General Case 362
15.3. Interior Gradient Bounds 369
15.4. Equations in Divergence Form 373
15.5. Selected Existence Theorems 380
15.6. Existence Theorems for Continuous Boundary Values 384
Notes 385
Problems 386
Table of Contents xiii
Chapter 16. Equations of Mean Curvature Type 388
16.1. HypersurfacesinR +1 388
16.2. Interior Gradient Bounds 401
16.3. Application to the Dirichlet Problem 407
16.4. Equations in Two Independent Variables 410
16.5. Quasiconformal Mappings 413
16.6. Graphs with Quasiconformal Gauss Map 423
16.7. Applications to Equations of Mean Curvature Type 429
16.8. Appendix: Elliptic Parametric Functionals 434
Notes 437
Problems 438
Chapter 17. Fully Nonlinear Equations 441
17.1. Maximum and Comparison Principles 443
17.2. The Method of Continuity 446
17.3. Equations in Two Variables 450
17.4. Holder Estimates for Second Derivatives 453
17.5. Dirichlet Problem for Uniformly Elliptic Equations 463
17.6. Second Derivative Estimates for Equations of
Monge Ampere Type 467
17.7. Dirichlet Problem for Equations of Monge Ampere Type . . . 471
17.8. Global Second Derivative Holder Estimates 476
17.9. Nonlinear Boundary Value Problems 481
Notes 485
Problems 487
Bibliography 490
Subject Index 507
Notation Index 512
|
| any_adam_object | 1 |
| author | Gilbarg, David 1918-2001 Trudinger, Neil S. 1942- |
| author_GND | (DE-588)122665805 (DE-588)122665813 |
| author_facet | Gilbarg, David 1918-2001 Trudinger, Neil S. 1942- |
| author_role | aut aut |
| author_sort | Gilbarg, David 1918-2001 |
| author_variant | d g dg n s t ns nst |
| building | Verbundindex |
| bvnumber | BV000351968 |
| classification_rvk | SK 560 SK 540 |
| ctrlnum | (OCoLC)9826367 (DE-599)BVBBV000351968 |
| dewey-full | 515.3/53 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3/53 |
| dewey-search | 515.3/53 |
| dewey-sort | 3515.3 253 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV000351968 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:12:46Z |
| institution | BVB |
| isbn | 354013025X 038713025X |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000218820 |
| oclc_num | 9826367 |
| open_access_boolean | |
| owner | DE-12 DE-384 DE-703 DE-739 DE-824 DE-29T DE-19 DE-BY-UBM DE-634 DE-11 DE-188 DE-83 |
| owner_facet | DE-12 DE-384 DE-703 DE-739 DE-824 DE-29T DE-19 DE-BY-UBM DE-634 DE-11 DE-188 DE-83 |
| physical | XIII, 513 S. |
| publishDate | 1983 |
| publishDateSearch | 1983 |
| publishDateSort | 1983 |
| publisher | Springer |
| record_format | marc |
| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Gilbarg, David 1918-2001 Verfasser (DE-588)122665805 aut Elliptic partial differential equations of second order David Gilbarg ; Neil S. Trudinger 2. ed. Berlin [u.a.] Springer 1983 XIII, 513 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 224 EDP inriac Equations différentielles elliptiques ram Equations du second degré ram Opérateurs elliptiques ram Équations différentielles elliptiques Équations différentielles elliptiques ram équation 2ème ordre inriac équation différentielle elliptique inriac équation elliptique inriac équation linéaire inriac équation quasilinéaire inriac Differential equations, Elliptic Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Ordnung 2 (DE-588)4350619-7 gnd rswk-swf Funktionalraum (DE-588)4155679-3 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Ordnung 2 (DE-588)4350619-7 s Funktionalraum (DE-588)4155679-3 s 1\p DE-604 Partielle Differentialgleichung (DE-588)4044779-0 s 2\p DE-604 Trudinger, Neil S. 1942- Verfasser (DE-588)122665813 aut Grundlehren der mathematischen Wissenschaften 224 (DE-604)BV000000395 224 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000218820&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 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Gilbarg, David 1918-2001 Trudinger, Neil S. 1942- Elliptic partial differential equations of second order Grundlehren der mathematischen Wissenschaften EDP inriac Equations différentielles elliptiques ram Equations du second degré ram Opérateurs elliptiques ram Équations différentielles elliptiques Équations différentielles elliptiques ram équation 2ème ordre inriac équation différentielle elliptique inriac équation elliptique inriac équation linéaire inriac équation quasilinéaire inriac Differential equations, Elliptic Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Ordnung 2 (DE-588)4350619-7 gnd Funktionalraum (DE-588)4155679-3 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4014485-9 (DE-588)4350619-7 (DE-588)4155679-3 |
| title | Elliptic partial differential equations of second order |
| title_auth | Elliptic partial differential equations of second order |
| title_exact_search | Elliptic partial differential equations of second order |
| title_full | Elliptic partial differential equations of second order David Gilbarg ; Neil S. Trudinger |
| title_fullStr | Elliptic partial differential equations of second order David Gilbarg ; Neil S. Trudinger |
| title_full_unstemmed | Elliptic partial differential equations of second order David Gilbarg ; Neil S. Trudinger |
| title_short | Elliptic partial differential equations of second order |
| title_sort | elliptic partial differential equations of second order |
| topic | EDP inriac Equations différentielles elliptiques ram Equations du second degré ram Opérateurs elliptiques ram Équations différentielles elliptiques Équations différentielles elliptiques ram équation 2ème ordre inriac équation différentielle elliptique inriac équation elliptique inriac équation linéaire inriac équation quasilinéaire inriac Differential equations, Elliptic Partielle Differentialgleichung (DE-588)4044779-0 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Ordnung 2 (DE-588)4350619-7 gnd Funktionalraum (DE-588)4155679-3 gnd |
| topic_facet | EDP Equations différentielles elliptiques Equations du second degré Opérateurs elliptiques Équations différentielles elliptiques équation 2ème ordre équation différentielle elliptique équation elliptique équation linéaire équation quasilinéaire Differential equations, Elliptic Partielle Differentialgleichung Elliptische Differentialgleichung Ordnung 2 Funktionalraum |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000218820&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
| work_keys_str_mv | AT gilbargdavid ellipticpartialdifferentialequationsofsecondorder AT trudingerneils ellipticpartialdifferentialequationsofsecondorder |