Regularity results for nonlinear elliptic systems and applications:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London
Springer
2002
|
| Schriftenreihe: | Applied mathematical sciences
151 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XII, 440 S. |
| ISBN: | 3540677569 |
Internformat
MARC
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| 245 | 1 | 0 | |a Regularity results for nonlinear elliptic systems and applications |c Alain Bensoussan ; Jens Frehse |
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| 650 | 4 | |a Differential equations, Elliptic -- Numerical solutions | |
| 650 | 4 | |a Differential equations, Nonlinear -- Numerical solutions | |
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Datensatz im Suchindex
| _version_ | 1804129229014564864 |
|---|---|
| adam_text | Titel: Regularity results for nonlinear elliptic systems and applications
Autor: Bensoussan, Alain
Jahr: 2002
Contents
Preface
v
1. General Technical Results................................ 1
1.1 Introduction........................................... 1
1.1.1 Function Spaces.................................. 1
1.1.2 Regularity of Domains............................ 10
1.1.3 Poincare Inequality............................... 12
1.1.4 Covering of Domains ............................. 18
1.2 Useful Techniques...................................... 25
1.2.1 Reverse Holder s Inequality........................ 25
1.2.2 Gehring s Result................................. 36
1.2.3 Hole-Filling Technique of Widman.................. 38
1.2.4 Inhomogeneous Hole-Filling........................ 40
1.3 Green Function ........................................ 44
1.3.1 Statement of Results.............................. 44
1.3.2 Proof of Theorem 1.26 ........................... 45
1.3.3 Estimates on logG............................... 46
1.3.4 Estimates on Positive and Negative Powers of G...... 49
1.3.5 Harnack s Inequality.............................. 52
1.3.6 Proof of Theorem 1.27............................ 57
2. General Regularity Results............................... 63
2.1 Introduction........................................... 63
2.2 Obtaining W1* Regularity.............................. 63
2.2.1 Linear Equations................................. 63
2.2.2 Nonlinear Problems............................... 66
2.3 Obtaining Cs Regularity................................ 70
2.3.1 L°° Bounds for Linear Problems ................... 70
2.3.2 Cs Regularity for Dirichlet Problems ............... 73
2.3.3 Cs Regularity for Linear Mixed Boundary Value
Problems........................................ 82
2.3.4 Cs Regularity in the Case n = 2.................... 85
2.4 Maximum Principle..................................... 87
2.4.1 Assumptions..................................... 87
viii Contents
2.4.2 Proof of Theorem 2.16............................ 88
2.5 More Regularity........................................ 89
2.5.1 From C 5 and W^ P0, po 2, to ff£c ............... 89
2.5.2 Using the Linear Theory of Regularity ............. 96
2.5.3 Full Regularity for a General Quasilinear Scalar
Equation........................................ 98
3. Nonlinear Elliptic Systems Arising from Stochastic Games 113
3.1 Stochastic Games Background ...........................113
3.1.1 Statement of the Problem and Results..............113
3.1.2 Bellman Equations...............................115
3.1.3 Verification Property .............................116
3.2 Introduction to the Analytic Part.........................118
3.3 Estimates in Sobolev spaces and in Cs....................120
3.3.1 Assumptions and Statement of Results..............120
3.3.2 Preliminaries ....................................122
3.3.3 Proof of Theorem 3.7 ............................125
3.4 Estimates in L°° .......................................127
3.4.1 Assumptions ....................................127
3.4.2 Statement of Results .............................128
3.5 Existence of Solutions...................................129
3.5.1 Setting of the Problem and Assumptions............129
3.5.2 Proof of Existence................................130
3.5.3 Existence of a Weak Solution......................132
3.6 Hamiltonians Arising from Games .......................133
3.6.1 Notation........................................133
3.6.2 Verification of the Assumptions for Holder Regularity 135
3.6.3 Verification of the Assumptions for the L°° Bound.... 136
3.7 The Case of Two Players with Different Coupling Terms in
the Payoffs............................................143
3.7.1 Description of the Model and Statement of Results ... 144
3.7.2 L°° Bounds .....................................145
3.7.3 Hi Bound.......................................150
4. Nonlinear Elliptic Systems Arising from Ergodic Control . 153
4.1 Introduction...........................................153
4.2 Assumptions and Statement of Results....................154
4.2.1 Assumptions on the Hamiltonians..................154
4.2.2 Statement of Results .............................156
4.3 Proof of Theorem 4.4...................................156
4.3.1 First Estimates ..................................156
4.3.2 Estimates on v% - u t .............................158
4.3.3 End of Proof of Theorem 4.4.......................161
4.4 Verification of the Assumptions..........................162
4.4.1 Notation........................................162
Contents ix
4.4.2 The Scalar Case..................................163
4.4.3 The General Case................................167
4.5 A Variant of Theorem 4.4...............................169
4.5.1 Statement of Results..............................169
4.5.2 Proof of Theorem 4.13............................170
4.6 Ergodic Problems in J? ...............................175
4.6.1 Presentation of the Problem.......................175
4.6.2 Existence Theorem for an Approximate Solution.....176
4.6.3 Proof of Theorem 4.17............................189
4.6.4 Growth at Infinity................................191
4.6.5 Uniqueness......................................192
5. Harmonic Mappings......................................197
5.1 Introduction........................................... 197
5.2 Extremals............................................. 198
5.3 Regularity............................................. 200
5.4 Hardy Spaces.......................................... 201
5.4.1 Basic Properties..................................201
5.4.2 Main Regularity Result in the Hardy Space..........204
5.5 Proof of Theorem 5.13..................................208
5.5.1 Continuity when n = 2............................208
5.5.2 Proof of (5.35) and (5.36) .........................216
5.5.3 Proof of (5.37) ..................................218
5.5.4 Atomic decomposition............................221
6. Nonlinear Elliptic Systems Arising
from the Theory of Semiconductors.......................229
6.1 Physical Background....................................229
6.2 Stationary Case Without Impact Ionization................230
6.2.1 Mathematical Setting.............................230
6.2.2 Proof of Theorem 6.1 ............................233
6.2.3 A Uniqueness Result .............................240
6.2.4 Local Regularity.................................245
6.3 Stationary Case with Impact Ionization...................246
6.3.1 Setting of the Model..............................246
6.3.2 Proof of Theorem 6.5.............................248
6.4 Impact Ionization Without Recombination................257
6.4.1 Statement of the Problem.........................257
6.4.2 Proof of Theorem 6.7.............................259
7. Stationary Navier-Stokes Equations......................265
7.1 Introduction...........................................265
7.2 Regularity of Maximum-Like Solutions ..................266
7.2.1 Setting of the Problem ...........................266
x Contents
7.2.2 Some Regularity Properties of Maximum-Like
Solutions ......................................267
7.2.3 The Navier-Stokes Inequality .....................273
7.2.4 Hole-Filling......................................275
7.2.5 Full Regularity ..................................279
7.3 Maximum Solutions and the NS Inequality ................280
7.3.1 Notation and Setup...............................280
7.3.2 Proof of Theorem 7.8.............................281
7.4 Existence of a Regular Solution for n 5..................283
7.4.1 Green Function Associated with Incompressible Flows 283
7.4.2 Approximation...................................288
7.4.3 Proof of Existence of a Maximum Solution for n 5 .. 289
7.5 Periodic Case: Existence of a Regular Solution for n 10-----291
7.5.1 Approximation...................................291
7.5.2 A Specific Green Function.........................292
7.5.3 Main Results ....................................295
8. Strongly Coupled Elliptic Systems........................299
8.1 Introduction...........................................299
8.2 H2^ and Meyers s Regularity Results.....................300
8.3 Holder Regularity......................................305
8.3.1 Preliminaries ....................................305
8.3.2 Representation Using Spherical Functions...........308
8.3.3 Statement of the Main Result......................311
8.3.4 Additional Remarks..............................317
8.3.5 Holder s Continuity up to the Boundary.............319
8.4 C1+a Regularity .......................................329
8.4.1 Auxiliary Inequalities.............................329
8.4.2 Main Result.....................................334
8.5 Almost Everywhere Regularity...........................338
8.5.1 Regularity on Neighborhoods of Lebesgue Points.....338
8.5.2 Proof of Theorem 8.22............................339
8.6 Regularity in the Uhlenbeck Case.........................343
8.6.1 Setting of the Problem............................343
8.6.2 Proof of Theorem 8.24............................344
8.7 Counterexamples.......................................348
8.8 Regularity for Mixed Boundary Value Systems.............352
8.8.1 Stating the Problem..............................352
8.8.2 Proof of Theorem 8.25............................354
8.8.3 Proof of Lemma 8.28 .............................359
8.8.4 Further Regularity ...............................364
8.8.5 Domain with a Corner. Mixed Boundary Conditions .. 369
8.8.6 Domain with a Corner. Dirichlet Boundary Conditions 371
Contents xi
9. Dual Approach to Nonlinear Elliptic Systems.............375
9.1 Introduction...........................................375
9.2 Preliminaries...........................................377
9.2.1 Notation........................................377
9.2.2 Properties of the Operators e(u) and Du............378
9.3 Elasticity Models.......................................379
9.3.1 Primal and Dual Problems ........................379
9.3.2 A Hybrid Model..................................380
9.4 H oc Theory for the Nonsymmetric Case...................381
9.4.1 Presentation of the Problem.......................381
9.4.2 H{oc Regularity..................................382
9.5 Hloc Theory for the Symmetric Case......................391
9.5.1 Presentation of the Problem.......................391
9.5.2 H^ Regularity..................................391
9.5.3 Reducing the Symmetric Case to the Nonsymmetric
Case............................................396
9.6 Lfoc Theory for the Nonsymmetric Uhlenbeck Case.........398
9.6.1 Setting of the Problem and Statement of Results.....398
9.6.2 Proof of Theorem 9.8.............................399
9.7 W^£ Theory for the Nonsymmetric Case..................401
9.7.1 Assumptions and Results..........................401
9.7.2 Proof of Theorem 9.9.............................402
9.8 C^5 Regularity for the Nonsymmetric Case...............405
9.8.1 Setting of the Problem and Statement of Results.....405
9.8.2 Preliminary Results...............................406
9.8.3 Proof of Theorem 9.10............................410
9.9 Cs Regularity on Neighborhoods of Lebesgue Points for the
Nonsymmetric Case.....................................413
9.9.1 Setting of the Problem and Statement of Results.....413
9.9.2 Proof of Theorem 9.11............................414
9.9.3 Additional Results in the Uhlenbeck Case...........418
10. Nonlinear Elliptic Systems Arising
from plasticity Theory....................................421
10.1 Introduction...........................................421
10.2 Description of Models...................................422
10.2.1 Spaces U((2), E(Q)...............................422
10.2.2 Hencky model...................................423
10.2.3 Norton-Hoff Model...............................424
10.2.4 Passing to the Limit..............................426
10.3 Estimates on the Displacement...........................427
10.3.1 The fj Derive from a Potential.....................427
10.3.2 Strict Interior Condition ..........................428
10.3.3 Constituent Law for the Hencky model..............429
10.4 H{oc Regularity ........................................430
xii Contents
10.4.1 Preliminaries ....................................430
10.4.2 Uniform Estimates and Main Regularity Result......432
References....................................................435
Index.........................................................441
|
| any_adam_object | 1 |
| author | Bensoussan, Alain 1940- Frehse, Jens 1943- |
| author_GND | (DE-588)13295947X (DE-588)106319930 |
| author_facet | Bensoussan, Alain 1940- Frehse, Jens 1943- |
| author_role | aut aut |
| author_sort | Bensoussan, Alain 1940- |
| author_variant | a b ab j f jf |
| building | Verbundindex |
| bvnumber | BV014330628 |
| callnumber-first | Q - Science |
| callnumber-label | QA1 QA377 |
| callnumber-raw | QA1 QA377.A647 no. 151 |
| callnumber-search | QA1 QA377.A647 no. 151 |
| callnumber-sort | QA 11 _Q A377 A647 NO 3151 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 SK 560 |
| ctrlnum | (OCoLC)49679386 (DE-599)BVBBV014330628 |
| dewey-full | 510S515/.35321 510 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 515 - Analysis |
| dewey-raw | 510 s 515/.353 21 510 515/.353 |
| dewey-search | 510 s 515/.353 21 510 515/.353 |
| dewey-sort | 3510 S 3515 3353 221 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV014330628 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T19:01:49Z |
| institution | BVB |
| isbn | 3540677569 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009831226 |
| oclc_num | 49679386 |
| open_access_boolean | |
| owner | DE-20 DE-384 DE-355 DE-BY-UBR DE-29T DE-703 DE-19 DE-BY-UBM DE-706 DE-521 DE-634 DE-83 DE-11 DE-188 |
| owner_facet | DE-20 DE-384 DE-355 DE-BY-UBR DE-29T DE-703 DE-19 DE-BY-UBM DE-706 DE-521 DE-634 DE-83 DE-11 DE-188 |
| physical | XII, 440 S. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Bensoussan, Alain 1940- Verfasser (DE-588)13295947X aut Regularity results for nonlinear elliptic systems and applications Alain Bensoussan ; Jens Frehse Berlin ; Heidelberg ; New York ; Barcelona ; Hong Kong ; London Springer 2002 XII, 440 S. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 151 Elliptische systemen gtt Niet-lineaire vergelijkingen gtt Équations différentielles elliptiques - Solutions numériques Équations différentielles non linéaires - Solutions numériques Differential equations, Elliptic -- Numerical solutions Differential equations, Nonlinear -- Numerical solutions Elliptisches System (DE-588)4121184-4 gnd rswk-swf Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd rswk-swf Regularität (DE-588)4049074-9 gnd rswk-swf Lösung Mathematik (DE-588)4120678-2 gnd rswk-swf Nichtlineares System (DE-588)4042110-7 gnd rswk-swf Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 s DE-604 Elliptisches System (DE-588)4121184-4 s Nichtlineares System (DE-588)4042110-7 s Lösung Mathematik (DE-588)4120678-2 s Regularität (DE-588)4049074-9 s Frehse, Jens 1943- Verfasser (DE-588)106319930 aut Applied mathematical sciences 151 (DE-604)BV000005274 151 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009831226&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Bensoussan, Alain 1940- Frehse, Jens 1943- Regularity results for nonlinear elliptic systems and applications Applied mathematical sciences Elliptische systemen gtt Niet-lineaire vergelijkingen gtt Équations différentielles elliptiques - Solutions numériques Équations différentielles non linéaires - Solutions numériques Differential equations, Elliptic -- Numerical solutions Differential equations, Nonlinear -- Numerical solutions Elliptisches System (DE-588)4121184-4 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd Regularität (DE-588)4049074-9 gnd Lösung Mathematik (DE-588)4120678-2 gnd Nichtlineares System (DE-588)4042110-7 gnd |
| subject_GND | (DE-588)4121184-4 (DE-588)4310554-3 (DE-588)4049074-9 (DE-588)4120678-2 (DE-588)4042110-7 |
| title | Regularity results for nonlinear elliptic systems and applications |
| title_auth | Regularity results for nonlinear elliptic systems and applications |
| title_exact_search | Regularity results for nonlinear elliptic systems and applications |
| title_full | Regularity results for nonlinear elliptic systems and applications Alain Bensoussan ; Jens Frehse |
| title_fullStr | Regularity results for nonlinear elliptic systems and applications Alain Bensoussan ; Jens Frehse |
| title_full_unstemmed | Regularity results for nonlinear elliptic systems and applications Alain Bensoussan ; Jens Frehse |
| title_short | Regularity results for nonlinear elliptic systems and applications |
| title_sort | regularity results for nonlinear elliptic systems and applications |
| topic | Elliptische systemen gtt Niet-lineaire vergelijkingen gtt Équations différentielles elliptiques - Solutions numériques Équations différentielles non linéaires - Solutions numériques Differential equations, Elliptic -- Numerical solutions Differential equations, Nonlinear -- Numerical solutions Elliptisches System (DE-588)4121184-4 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd Regularität (DE-588)4049074-9 gnd Lösung Mathematik (DE-588)4120678-2 gnd Nichtlineares System (DE-588)4042110-7 gnd |
| topic_facet | Elliptische systemen Niet-lineaire vergelijkingen Équations différentielles elliptiques - Solutions numériques Équations différentielles non linéaires - Solutions numériques Differential equations, Elliptic -- Numerical solutions Differential equations, Nonlinear -- Numerical solutions Elliptisches System Nichtlineare elliptische Differentialgleichung Regularität Lösung Mathematik Nichtlineares System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009831226&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT bensoussanalain regularityresultsfornonlinearellipticsystemsandapplications AT frehsejens regularityresultsfornonlinearellipticsystemsandapplications |