Elliptic equations: an introductory course
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Basel [u.a.]
Birkhäuser
2009
|
| Schriftenreihe: | Birkhäuser advanced texts
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | VIII, 288 S. graph. Darst. |
| ISBN: | 9783764399818 9783764399825 |
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Datensatz im Suchindex
| _version_ | 1804138532858494976 |
|---|---|
| adam_text | CONTENTS PREFACE . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . IX PART I BASIC TECHNIQUES 1 HILBERT SPACE
TECHNIQUES 1.1 THE PROJECTION ON A CLOSED CONVEX SET . . . . . . . . . .
. . . . . . 3 1.2 THE RIESZ REPRESENTATION THEOREM . . . . . . . . . . .
. . . . . . . 6 1.3 THE LAX*MILGRAM THEOREM . . . . . . . . . . . . . .
. . . . . . . . 8 1.4 CONVERGENCE TECHNIQUES . . . . . . . . . . . . . .
. . . . . . . . . 10 EXERCISES . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 11 2 A SURVEY OF ESSENTIAL ANALYSIS 2.1 L P
-TECHNIQUES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 INTRODUCTION TO DISTRIBUTIONS . . . . . . . . . . . . . . . . . . .
. . 18 2.3 SOBOLEV SPACES . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 22 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 32 3 WEAK FORMULATION OF ELLIPTIC PROBLEMS 3.1
MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 3.2 THE WEAK FORMULATION . . . . . . . . . . . . . . . . . . . . . .
. . 38 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41 4 ELLIPTIC PROBLEMS IN DIVERGENCE FORM 4.1 WEAK
FORMULATION . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.2
THE WEAK MAXIMUM PRINCIPLE . . . . . . . . . . . . . . . . . . . . 49
4.3 INHOMOGENEOUS PROBLEMS . . . . . . . . . . . . . . . . . . . . . . .
53 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 54 5 SINGULAR PERTURBATION PROBLEMS 5.1 A PROTOTYPE OF A
SINGULAR PERTURBATION PROBLEM . . . . . . . . . . 57 5.2 ANISOTROPIC
SINGULAR PERTURBATION PROBLEMS . . . . . . . . . . . . 61 EXERCISES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 VI
CONTENTS 6 PROBLEMS IN LARGE CYLINDERS 6.1 A MODEL PROBLEM . . . . . . .
. . . . . . . . . . . . . . . . . . . . 73 6.2 ANOTHER TYPE OF
CONVERGENCE . . . . . . . . . . . . . . . . . . . . . 79 6.3 THE GENERAL
CASE . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 6.4 AN
APPLICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 89 7 PERIODIC PROBLEMS 7.1 A GENERAL THEORY . . . . . . . . . .
. . . . . . . . . . . . . . . . . 93 7.2 SOME ADDITIONAL REMARKS . . . .
. . . . . . . . . . . . . . . . . . . 101 EXERCISES . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 103 8 HOMOGENIZATION
8.1 MORE ON PERIODIC FUNCTIONS . . . . . . . . . . . . . . . . . . . . .
. 106 8.2 HOMOGENIZATION OF ELLIPTIC EQUATIONS . . . . . . . . . . . . .
. . . 109 8.2.1 THE ONE-DIMENSIONAL CASE . . . . . . . . . . . . . . . .
. . 109 8.2.2 THE N -DIMENSIONAL CASE . . . . . . . . . . . . . . . . .
. . 112 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 119 9 EIGENVALUES 9.1 THE ONE-DIMENSIONAL CASE . . . . .
. . . . . . . . . . . . . . . . . 121 9.2 THE HIGHER-DIMENSIONAL CASE .
. . . . . . . . . . . . . . . . . . . . 123 9.3 AN APPLICATION . . . . .
. . . . . . . . . . . . . . . . . . . . . . . 127 EXERCISES . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 10
NUMERICAL COMPUTATIONS 10.1 THE FINITE DIFFERENCE METHOD . . . . . . . .
. . . . . . . . . . . . . 129 10.2 THE FINITE ELEMENT METHOD . . . . . .
. . . . . . . . . . . . . . . . 135 EXERCISES . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 147 PART II MORE ADVANCED
THEORY 11 NONLINEAR PROBLEMS 11.1 MONOTONE METHODS . . . . . . . . . . .
. . . . . . . . . . . . . . . 153 11.2 QUASILINEAR EQUATIONS . . . . . .
. . . . . . . . . . . . . . . . . . . 160 11.3 NONLOCAL PROBLEMS . . . .
. . . . . . . . . . . . . . . . . . . . . . 166 11.4 VARIATIONAL
INEQUALITIES . . . . . . . . . . . . . . . . . . . . . . . . 170
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 174 CONTENTS VII 12 L * -ESTIMATES 12.1 SOME SIMPLE CASES . . .
. . . . . . . . . . . . . . . . . . . . . . . 177 12.2 A MORE INVOLVED
ESTIMATE . . . . . . . . . . . . . . . . . . . . . . 180 12.3 THE
SOBOLEV*GAGLIARDO*NIRENBERG INEQUALITY . . . . . . . . . . . 183 12.4
THE MAXIMUM PRINCIPLE ON SMALL DOMAINS . . . . . . . . . . . . . 187
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 188 13 LINEAR ELLIPTIC SYSTEMS 13.1 THE GENERAL FRAMEWORK . . .
. . . . . . . . . . . . . . . . . . . . . 191 13.2 SOME EXAMPLES . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 197 EXERCISES . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 14 THE
STATIONARY NAVIER*STOKES SYSTEM 14.1 INTRODUCTION . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 203 14.2 EXISTENCE AND UNIQUENESS
RESULT . . . . . . . . . . . . . . . . . . . 205 EXERCISES . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 15 SOME MORE
SPACES 15.1 MOTIVATION . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 211 15.2 ESSENTIAL FEATURES OF THE SOBOLEV SPACES W K,P . .
. . . . . . . . . 212 15.3 AN APPLICATION . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 215 EXERCISES . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 216 16 REGULARITY THEORY 16.1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
217 16.2 THE TRANSLATION METHOD . . . . . . . . . . . . . . . . . . . .
. . . 221 16.3 REGULARITY OF FUNCTIONS IN SOBOLEV SPACES . . . . . . . .
. . . . . . 224 16.4 THE BOOTSTRAP TECHNIQUE . . . . . . . . . . . . . .
. . . . . . . . . 227 EXERCISES . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 229 17 THE P -LAPLACE EQUATION 17.1 A
MINIMIZATION TECHNIQUE . . . . . . . . . . . . . . . . . . . . . . 231
17.2 A WEAK MAXIMUM PRINCIPLE AND ITS CONSEQUENCES . . . . . . . . . 237
17.3 A GENERALIZATION OF THE LAX*MILGRAM THEOREM . . . . . . . . . . .
239 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 245 18 THE STRONG MAXIMUM PRINCIPLE 18.1 A FIRST VERSION OF
THE MAXIMUM PRINCIPLE . . . . . . . . . . . . . . 247 18.2 THE HOPF
MAXIMUM PRINCIPLE . . . . . . . . . . . . . . . . . . . . 252 18.3
APPLICATION: THE MOVING PLANE TECHNIQUE . . . . . . . . . . . . . . 255
EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 259 VIII CONTENTS 19 PROBLEMS IN THE WHOLE SPACE 19.1 THE
HARMONIC FUNCTIONS, LIOUVILLE THEOREM . . . . . . . . . . . . . 261 19.2
THE SCHR¨ ODINGER EQUATION . . . . . . . . . . . . . . . . . . . . . .
267 EXERCISES . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 273 APPENDIX: FIXED POINT THEOREMS A.1 THE BROUWER FIXED
POINT THEOREM . . . . . . . . . . . . . . . . . . 275 A.2 THE SCHAUDER
FIXED POINT THEOREM . . . . . . . . . . . . . . . . . 279 EXERCISES . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 281 INDEX . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . 287
|
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| illustrated | Illustrated |
| indexdate | 2024-07-09T21:29:42Z |
| institution | BVB |
| isbn | 9783764399818 9783764399825 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017058731 |
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| physical | VIII, 288 S. graph. Darst. |
| publishDate | 2009 |
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| publisher | Birkhäuser |
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| spelling | Chipot, Michel 1949- Verfasser (DE-588)110320999 aut Elliptic equations an introductory course Michel Chipot Basel [u.a.] Birkhäuser 2009 VIII, 288 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Birkhäuser advanced texts Elliptische Differentialgleichung - Lehrbuch Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s DE-604 SWBplus Fremddatenuebernahme application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017058731&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chipot, Michel 1949- Elliptic equations an introductory course Elliptische Differentialgleichung - Lehrbuch Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| subject_GND | (DE-588)4014485-9 |
| title | Elliptic equations an introductory course |
| title_auth | Elliptic equations an introductory course |
| title_exact_search | Elliptic equations an introductory course |
| title_full | Elliptic equations an introductory course Michel Chipot |
| title_fullStr | Elliptic equations an introductory course Michel Chipot |
| title_full_unstemmed | Elliptic equations an introductory course Michel Chipot |
| title_short | Elliptic equations |
| title_sort | elliptic equations an introductory course |
| title_sub | an introductory course |
| topic | Elliptische Differentialgleichung - Lehrbuch Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd |
| topic_facet | Elliptische Differentialgleichung - Lehrbuch Differential equations, Elliptic Elliptische Differentialgleichung |
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