Elliptic differential equations and obstacle problems:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Plenum Pr.
1987
|
| Series: | The university series in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XVI, 353 S. |
| ISBN: | 0306424487 |
Staff View
MARC
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| 100 | 1 | |a Troianiello, Giovanni M. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Elliptic differential equations and obstacle problems |c Giovanni Maria Troianiello |
| 264 | 1 | |a New York [u.a.] |b Plenum Pr. |c 1987 | |
| 300 | |a XVI, 353 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 0 | |a The university series in mathematics | |
| 650 | 4 | |a Calculus of variations | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 4 | |a Variational inequalities (Mathematics) | |
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Record in the Search Index
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| adam_text | Contents
Glossary of Basic Notations xv
1. Function Spaces 1
1.1. Preliminaries from Functional Analysis 2
1.1.1. Banach and Hilbert Spaces 2
1.1.2. Fixed Points and Compact Operators 6
1.2. Various Spaces of Smooth Functions 8
1.2.1. Ck and Ck d Spaces 8
1.2.2. Extensions 11
1.2.3. Traces 15
1.3. Lebesgue Spaces 16
1.3.1. LP Spaces over Q 17
1.3.2. Approximation by Convolution in C°, C0(5, D . . . 19
1.3.3. LP Spaces over r 24
1.4. Morrey, John Nirenberg, and Campanato Spaces 28
1.4.1. Definition and Basic Properties 28
1.4.2. Equivalent Norms and Multipliers 31
1.5. Sobolev Spaces 38
1.5.1. Distributional Derivatives 39
1.5.2. Difference Quotients 42
1.5.3. Hkp Spaces: Definitions and First Properties .... 44
1.5.4. Density Results 47
1.5.5. Changes of Variables and Extensions 50
1.6. Continuous and Compact Imbeddings of Sobolev Spaces . . 54
1.6.1. Sobolev Inequalities I 54
xi
xii Contents
1.6.2. Rellich s Theorem with Some Applications 58
1.6.3. Sobolev Inequalities II 61
1.7. Hok P Spaces and Trace Spaces 64
1.7.1. Hok P(Q) Spaces 64
1.7.2. HQ^(Q u F) Spaces 67
1.7.3. Boundary Values and H^ ^r) Spaces 69
1.7.4. Supplementary Results 74
1.8. Inequalities and Lattice Properties 76
1.8.1. Some Notions from the Abstract Theory of Ordered
Linear Spaces 76
1.8.2. Inequalities and Lattice Properties in Function Spaces
over Q 79
1.8.3. Boundary Inequalities 82
Problems 85
2. The Variational Theory of Elliptic Boundary Value Problems ... 89
2.1. Abstract Existence and Uniqueness Results 91
2.2. Variational Formulation of Boundary Value Problems ... 94
2.2.1. Bilinear Forms 94
2.2.2. The Weak Maximum Principle 96
2.2.3. Interpretation of Solutions 99
2.3. U Regularity of Solutions 102
2.4. The De Giorgi Nash Theorem 110
2.4.1. Pointwise Bounds on Subsolutions 110
2.4.2. Holder Continuity of Solutions 113
2.4.3. L2 Regularity of First Derivatives 117
2.5. Hk Regularity by the Method of Difference Quotients ... 125
2.5.1. Regularity in the Interior 125
2.5.2. Boundary and Global Regularity 129
2.6. Interior Regularity for Nonlinear Equations 133
2.6.1. Local Boundedness 134
2.6.2. H2 Regularity 138
2.6.3. Hlo° and Ch d Regularity 140
Problems 143
3. Hk* and C* * Theory 145
3.1. Estimates on Spheres 146
3.1.1. Homogeneous Equations with Constant Coefficients . 146
3.1.2. Nonhomogeneous Equations with Variable Coefficients 150
Contents xiii
3.2. Interior L2 Regularity of Derivatives 152
3.2.1. Regularity of First Derivatives 152
3.2.2. Regularity of Second Derivatives 157
3.3. Interior LP Regularity of Derivatives 160
3.4. Estimates on Hemispheres 165
3.4.1. Homogeneous Equations with Constant Coefficients . 165
3.4.2. Nonhomogeneous Equations with Variable Coefficients 170
3.5. Boundary and Global Regularity of Derivatives 172
3.5.1. L2^ Regularity near the Boundary 172
3.5.2. LP Regularity near the Boundary 177
3.5.3. Global Regularity 179
3.6. A priori Estimates on Solutions to Nonvariational Boundary
Value Problems 180
3.6.1. The Case of Smooth Coefficients 180
3.6.2. The General Case 183
3.7. Unique Solvability of Nonvariational Boundary Value
Problems 187
3.7.1. Regularity of Solutions 187
3.7.2. Maximum Principles 189
3.7.3. Existence and Uniqueness 194
3.8. The Marcinkiewicz Theorem and the John Nirenberg Lemma 196
Problems 201
4. Variational Inequalities 205
4.1. Minimization of Convex Functionals, and Variational
Inequalities for Linear Operators 206
4.1.1. A Class of Minimum Problems 206
4.1.2. Variational Inequalities 209
4.2. Variational Inequalities for Nonlinear Operators 215
4.2.1. Monotone and Pseudomonotone Operators 216
4.2.2. Existence and Approximation of Solutions 222
4.3. Variational Inequalities in Sobolev Spaces 227
4.3.1. Convex Sets 227
4.3.2. Bilinear Forms and Nonlinear Operators 228
4.3.3. Interpretation of Solutions 237
4.4. Existence and Uniqueness Results for a Class of Noncoercive
Bilinear Forms 241
4.4.1. Unilateral Variational Inequalities 242
4.4.2. Bilateral Variational Inequalities 247
xiv Contents
4.5. Lewy Stampacchia Inequalities and Applications
to Regularity 249
4.6. Further H** Regularity 255
4.6.1. H2°° Regularity 255
4.6.2. H2 Regularity up to F under General Conditions . . . 262
4.7. Regularity in Morrey and Campanato Spaces 266
4.7.1. The Case of Continuous Leading Coefficients .... 266
4.7.2. The Case of Holderian Leading Coefficients .... 270
4.7.3. The Case of Discontinuous Leading Coefficients . . 272
4.8. Lipschitz Regularity by the Penalty Method 275
4.9. Problems Involving Natural Growth of Nonlinear Terms . . 279
Problems 288
5. Nonyariational Obstacle Problems 291
5.1. Obstacle Problems for Linear Operators 292
5.1.1. Bilateral Problems 292
5.1.2. Unilateral Problems 294
5.1.3. An Approximation Result 299
5.1.4. Systems of Unilateral Problems 301
5.2. Differential Inequalities 305
5.2.1. Interpolation Results 306
5.2.2. A Global Bound 308
5.2.3. A Local Bound 312
5.3. Obstacle Problems for Nonlinear Operators 316
5.3.1. Existence 316
5.3.2. Uniqueness 321
5.4. Generalized Solutions and Implicit Unilateral Problems . . 323
5.4.1. Generalized Solutions 323
5.4.2. Implicit Unilateral Problems 326
5.4.3. The Implicit Unilateral Problem of Stochastic
Impulse Control 329
Problems 333
Bibliographical Notes 335
References 341
Index of Special Symbols and Abbreviations
Index 351
|
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| isbn | 0306424487 |
| language | English |
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| spelling | Troianiello, Giovanni M. Verfasser aut Elliptic differential equations and obstacle problems Giovanni Maria Troianiello New York [u.a.] Plenum Pr. 1987 XVI, 353 S. txt rdacontent n rdamedia nc rdacarrier The university series in mathematics Calculus of variations Differential equations, Elliptic Variational inequalities (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Variationsungleichung (DE-588)4187420-1 gnd rswk-swf Hindernisproblem (DE-588)4159901-9 gnd rswk-swf Elliptischer Differentialoperator (DE-588)4140057-4 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Variationsungleichung (DE-588)4187420-1 s DE-604 Elliptischer Differentialoperator (DE-588)4140057-4 s Hindernisproblem (DE-588)4159901-9 s Variationsrechnung (DE-588)4062355-5 s HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000464180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Troianiello, Giovanni M. Elliptic differential equations and obstacle problems Calculus of variations Differential equations, Elliptic Variational inequalities (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd Variationsrechnung (DE-588)4062355-5 gnd Variationsungleichung (DE-588)4187420-1 gnd Hindernisproblem (DE-588)4159901-9 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4062355-5 (DE-588)4187420-1 (DE-588)4159901-9 (DE-588)4140057-4 |
| title | Elliptic differential equations and obstacle problems |
| title_auth | Elliptic differential equations and obstacle problems |
| title_exact_search | Elliptic differential equations and obstacle problems |
| title_full | Elliptic differential equations and obstacle problems Giovanni Maria Troianiello |
| title_fullStr | Elliptic differential equations and obstacle problems Giovanni Maria Troianiello |
| title_full_unstemmed | Elliptic differential equations and obstacle problems Giovanni Maria Troianiello |
| title_short | Elliptic differential equations and obstacle problems |
| title_sort | elliptic differential equations and obstacle problems |
| topic | Calculus of variations Differential equations, Elliptic Variational inequalities (Mathematics) Elliptische Differentialgleichung (DE-588)4014485-9 gnd Variationsrechnung (DE-588)4062355-5 gnd Variationsungleichung (DE-588)4187420-1 gnd Hindernisproblem (DE-588)4159901-9 gnd Elliptischer Differentialoperator (DE-588)4140057-4 gnd |
| topic_facet | Calculus of variations Differential equations, Elliptic Variational inequalities (Mathematics) Elliptische Differentialgleichung Variationsrechnung Variationsungleichung Hindernisproblem Elliptischer Differentialoperator |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000464180&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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