Direct methods in the theory of elliptic equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin [u.a.]
Springer
2012
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| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XVI, 372 S. graph. Darst. |
| ISBN: | 9783642104541 |
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| 240 | 1 | 0 | |a Les méthodes directes en théorie des équations elliptiques |
| 245 | 1 | 0 | |a Direct methods in the theory of elliptic equations |c Jindřich Nečas |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2012 | |
| 300 | |a XVI, 372 S. |b graph. Darst. | ||
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Record in the Search Index
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IMAGE 1
CONTENTS
1 ELEMENTARY DESCRIPTION OF PRINCIPAL RESULTS 1
1.1 BEPPO LEVI SPACES 1
1.1.1 DEFINITION OF W K ' 2 1
1.1.2 EQUIVALENT NORMS 3
1.1.3 CONCEPT OF A TRACE 4
1.1.4 THE POINCARE INEQUALITY 6
1.1.5 RELLICH'S THEOREM 7
1.1.6 THE GENERALIZED POINCARE INEQUALITY 9
1.1.7 THE QUOTIENT SPACES 10
1.1.8 OTHER EQUIVALENT NORMS 11
1.1.9 AN IMBEDDING THEOREM 12
1.2 BOUNDARY VALUE PROBLEMS FOR ELLIPTIC OPERATORS 14
1.2.1 ELLIPTIC OPERATORS 14
1.2.2 DECOMPOSITION OF OPERATORS 16
1.2.3 THE BOUNDARY OPERATORS 17
1.2.4 GREEN'S FORMULA 18
1.2.5 SESQUILINEAR FORMS 21
1.2.6 BOUNDARY VALUE PROBLEMS 23
1.2.7 EXAMPLES 24
1.3 THE F-ELLIPTICITY, EXISTENCE AND UNIQUENESS OF THE SOLUTION 29 1.3.1
THE LAX-MILGRAM LEMMA 29
1.3.2 SOLVING THE BOUNDARY VALUE PROBLEM 30
1.3.3 THE CASE OF QUOTIENT SPACES 31
1.3.4 CONDITIONS OF V-ELLIPTICITY 33
1.3.5 NONSTABLE BOUNDARY CONDITIONS 37
1.3.6 ORTHOGONAL PROJECTIONS 37
1.4 THE RITZ, GALERKIN, AND LEAST SQUARES METHODS 38
1.4.1 THE VARIATIONAL METHOD 38
1.4.2 THE GALERKIN METHOD 39
1.4.3 THE LEAST SQUARES METHOD 41
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/1000033643
DIGITALISIERT DURCH
IMAGE 2
XII CONTENTS
1.5 BASIC NOTIONS FROM THE SPECTRAL THEORY 42
1.5.1 EIGENVALUES AND EIGENFUNCTIONS, THE FREDHOLM ALTERNATIVE 42
1.5.2 EIGENVALUES AND EIGENFUNCTIONS, THE FREDHOLM ALTERNATIVE
(CONTINUATION) 43
1.5.3 THE GAERDING INEQUALITY 45
2 THE SPACES W K P 49
2.1 DEFINITIONS AND AUXILIARY THEOREMS 49
2.1.1 CLASSIFICATION OF DOMAINS, PSEUDOTOPOLOGY IN C^(Q) 49 2.1.2 THE
SPACE L P (Q), MEAN CONTINUITY 51
2.1.3 THE REGULARIZING OPERATOR 52
2.1.4 COMPACTNESS CONDITION 53
2.2 THE SPACES W K 'P(Q) 54
2.2.1 A PROPERTY OF THE REGULARIZING OPERATOR 54
2.2.2 THE ABSOLUTE CONTINUITY 55
2.2.3 THE SPACES W K P(Q) 56
2.2.4 THE SPACES W K P (C2) (CONTINUATION) 58
2.2.5 THE SPACES W^ P (2) 59
2.3 IMBEDDING THEOREMS 60
2.3.1 THE LIPSCHITZ TRANSFORM 60
2.3.2 DENSITY OF C ( SS) IN W K P(Q) 61
2.3.3 THE GAGLIARDO LEMMA 62
2.3.4 THE SOBOLEV IMBEDDING THEOREMS 63
2.3.5 THE SOBOLEV IMBEDDING THEOREMS (CONTINUATION) 66 2.3.6 EXTENSION,
THE NIKOLSKII METHOD 69
2.3.7 EXTENSION, THE CALDERON METHOD 72
2.3.8 THE SPACES W K 'P(2), K NON-INTEGER 76
2.4 THE PROBLEM OF TRACES 77
2.4.1 LEMMAS 77
2.4.2 IMBEDDING THEOREMS 79
2.4.3 TWO TRACE THEOREMS 82
2.4.4 SOME OTHER PROPERTIES OF TRACES 83
2.5 THE PROBLEM OF TRACES (CONTINUATION) 86
2.5.1 APPLICATION OF THE FOURIER TRANSFORM 86
2.5.2 LEMMAS BASED ON THE HARDY INEQUALITY 90
2.5.3 IMBEDDING THEOREMS, APPLICATION OF THE SPACES W L ~ L L P ' P
{DL) 93
2.5.4 IMBEDDING THEOREMS, APPLICATION OF THE SPACES W X - X IP'P(DQ.)
(CONTINUATION) 93
2.5.5 A LEMMA 96
2.5.6 THE CONVERSE THEOREM 98
2.5.7 THE CONVERSE THEOREM (CONTINUATION) 99
2.5.8 REMARKS 101
IMAGE 3
CONTENTS
2.6 COMPACTNESS 102
2.6.1 THE KONDRASHOV THEOREM 102
2.6.2 TRACES 103
2.6.3 THE LIONS LEMMA, ANOTHER THEOREM OF COMPACTNESS 104 2.7 QUOTIENT
SPACES, EQUIVALENT NORMS 106
2.7.1 EQUIVALENT NORMS 106
2.7.2 QUOTIENT SPACES 108
2.7.3 THE SPACES V ^ F L) 109
2.7.4 NIKODYM DOMAINS I LL
EXISTENCE, UNIQUENESS AND FUNDAMENTAL PROPERTIES OF SOLUTIONS OF
BOUNDARY VALUE PROBLEMS 115
3.1 THE BOUNDARY INTEGRAL, GREEN'S FORMULA 115
3.1.1 THE BOUNDARY INTEGRAL 115
3.1.2 GREEN'S FORMULA 117
3.2 FORMULATION OF THE PROBLEM. EXISTENCE AND UNIQUENESS OF THE SOLUTION
119
3.2.1 SESQUILINEAR BOUNDARY FORMS 119
3.2.2 SESQUILINEAR BOUNDARY FORMS (CONTINUATION) 121 3.2.3 BOUNDARY
VALUE PROBLEMS 122
3.2.4 REMARKS 126
3.2.5 THE DIFFERENTIAL OPERATORS 127
3.3 THE FREDHOLM ALTERNATIVE 130
3.3.1 REMARKS 132
3.4 THE V-ELLIPTICITY 133
3.4.1 COERCIVITY 133
3.4.2 THE ARONSZAJN THEOREM 136
3.4.3 STRONGLY ELLIPTIC OPERATORS 138
3.4.4 ALGEBRAIC CONDITIONS FOR THE W*' 2 (I2)-ELLIPTICITY 141 3.5 THE
V-ELLIPTICITY OF FORMS J A AVAUAX 144
3.5.1 DEFINITION 144
3.5.2 THE FUNDAMENTAL SOLUTION 145
3.5.3 ALEMMA 147
3.5.4 A PRIORI ESTIMATES IN E+ 148
3.5.5 PROPERLY ELLIPTIC OPERATORS 154
3.5.6 PROPERLY ELLIPTIC OPERATORS (CONTINUATION) 159
3.6 CONTINUOUS DEPENDENCE ON THE DATA 160
3.6.1 DEPENDENCE ON THE COEFFICIENTS 160
3.6.2 DEPENDENCE ON THE COEFFICIENTS (CONTINUATION) 162 3.6.3 THE
SINGULAR CASE 164
3.6.4 DEPENDENCE ON THE SPACE V 168
3.6.5 DEPENDENCE ON THE SPACE V (CONTINUATION) 170
3.6.6 DEPENDENCE ON THE DOMAIN, THE DIRICHLET PROBLEM 172 3.6.7 THE
GENERAL CASE 173
3.6.8 DEPENDENCE ON THE DOMAIN, ANOTHER METHOD 179
IMAGE 4
CONTENTS
3.7 ELLIPTIC SYSTEMS 181
3.7.1 ELLIPTIC SYSTEMS AND SESQUILINEAR FORMS 182
3.7.2 BOUNDARY VALUE PROBLEMS 183
3.7.3 STRONGLY ELLIPTIC SYSTEMS 184
3.7.4 ALGEBRAICALLY COMPLETE, FORMALLY POSITIVE FORMS 185 3.7.5 EXAMPLES
192
REGULARITY OF THE SOLUTION 197
4.1 INTERIOR REGULARITY 198
4.1.1 REGULARITY OF THE WEAK SOLUTION 198
4.1.2 REGULARITY OF THE VERY WEAK SOLUTION 201
4.2 REGULARITY OF THE SOLUTION IN THE NEIGHBORHOOD OF THE BOUNDARY.
203 4.2.1 THE SECOND ORDER OPERATOR 203
4.2.2 THE REGULARIZABLE PROBLEM 204
4.2.3 LEMMAS 206
4.2.4 LEMMAS (CONTINUATION) 210
4.2.5 A MODIFICATION OF LIONS'LEMMA 211
4.2.6 A FUNDAMENTAL LEMMA 214
4.2.7 REGULARITY OF THE SOLUTION IN A NEIGHBORHOOD OFTHEBOUNDARY 218
4.2.8 STRONG SOLUTIONS 220
4.2.9 LOCAL REGULARITY IN A NEIGHBORHOOD OF THE BOUNDARY 223 4.2.10
DEPENDENCE OF THE SOLUTION ON THE COEFFICIENTS 223 4.3 BOUNDARY VALUE
PROBLEMS FOR PROPERLY ELLIPTIC OPERATORS 225 4.3.1 THE OPERATOR A
PROPERLY ELLIPTIC, B S COVER A 225
4.3.2 AN EXISTENCE THEOREM 227
4.3.3 DEPENDENCE WITH RESPECT TO A PARAMETER 230
4.4 VERY WEAK SOLUTIONS OF BOUNDARY VALUE PROBLEMS 231
4.4.1 VERY WEAK SOLUTIONS, THE HOMOGENEOUS CASE 231 4.4.2 REGULARITY OF
THE SOLUTION 232
4.4.3 VERY WEAK SOLUTIONS (CONTINUATION) 233
4.4.4 THE GREEN KERNEL 235
4.4.5 THE GREEN OPERATOR AND THE GREEN KERNEL 237
4.5 VERY WEAK SOLUTIONS (CONTINUATION) 239
4.5.1 VERY WEAK SOLUTIONS, THE NONHOMOGENEOUS CASE 239 4.5.2 VERY WEAK
SOLUTIONS, THE NONHOMOGENEOUS CASE (CONTINUATION) 242
4.5.3 THE GREEN AND POISSON KERNELS 245
APPLICATIONS OF RELLICH'S EQUALITIES AND THEIR GENERALIZATIONS TO
BOUNDARY VALUE PROBLEMS 247
5. 1 THE RELLICH EQUALITY FOR A SECOND ORDER EQUATION 247
5.1.1 THE RELLICH EQUALITY 247
5.1.2 LEMMAS, REGULARITY OF THE DIRICHLET PROBLEM 248
5.1.3 VERY WEAK SOLUTIONS 253
IMAGE 5
CONTENTS XV
5.2 THE NEUMANN AND NEWTON PROBLEMS 256
5.2.1 LEMMAS, REGULARITY OF THE SOLUTION 256
5.2.2 LEMMAS, REGULARITY OF THE SOLUTION (CONTINUATION) 259 5.2.3 VERY
WEAK SOLUTIONS 259
5.2.4 UNIQUENESS THEOREMS 260
5.2.5 VERY WEAK SOLUTIONS (CONTINUATION) 261
5.3 SECOND ORDER STRONGLY ELLIPTIC SYSTEMS 262
5.3.1 DEFINITIONS 262
5.3.2 REGULARITY OFTHE SOLUTION, I2 SMOOTH 263
5.3.3 THE RELLICH EQUALITY 264
5.3.4 REGULARITY OF THE SOLUTION, I2 NON-SMOOTH 266
5.3.5 VERY WEAK SOLUTIONS 267
5.4 A FOURTH ORDER EQUATION, THE DIRICHLET PROBLEM 268
5.4.1 DEFINITION 268
5.4.2 THE SECOND ORDER RELLICH INEQUALITY 269
5.4.3 THE THIRD ORDER RELLICH INEQUALITY 271
5.4.4 DEPENDENCE ON THE DOMAIN 272
5.4.5 A DENSITY LEMMA 274
5.4.6 REGULARITY OF THE SOLUTION 276
5.4.7 VERY WEAK SOLUTIONS 278
6 BOUNDARY VALUE PROBLEMS IN WEIGHTED SOBOLEV SPACES 281 6.1 A SECOND
ORDER EQUATION, REGULARITY OF SOLUTION 281
6.1.1 THE CASE OF DQ REGULAR 281
6.1.2 THE VERY WEAK SOLUTION 284
6.1.3 THE CASE D2 NON-SMOOTH 287
6.2 THE DIRICHLET PROBLEM AND SPACES W*' P 288
6.2.1 DENSITY THEOREM 288
6.2.2 THE TRACE PROBLEM 290
6.2.3 SOME IMBEDDING THEOREMS 292
6.2.4 THE DIRICHLET PROBLEM, VERY WEAK SOLUTION 294
6.3 SESQUILINEAR FORMS ON W^ T (Q)XW^' 2 (I2) 300
6.3.1 THESS,# 2 -ELLIPTICITY 300
6.3.2 ELLIPTICITY OF SESQUILINEAR FORMS FOR W'*. A , W A 301
6.3.3 THE DIRICHLET PROBLEM '. 309
6.3.4 THE NEUMANN PROBLEM AND OTHER PROBLEMS 310
6.3.5 AN IMBEDDING THEOREM FOR W^ P (Q) 311
6.3.6 CONCLUDING REMARKS 316
7 REGULARITY OF THE SOLUTION FOR NON-SMOOTH COEFFICIENTS AND NON-REGULAR
DOMAINS 319
7.1 SECOND ORDER OPERATORS 320
7.1.1 AUXILIARY LEMMAS 320
7.1.2 FUNDAMENTAL LEMMAS 322
IMAGE 6
XVI CONTENTS
7.1.3 REGULARITY OF THE SOLUTION OF THE BOUNDARY VALUE PROBLEM 325
7.1.4 THECASE 1/JV \/P 326
7.1.5 REGULARITY OF THE SOLUTION 328
7.1.6 THE DUAL METHOD 329
7.2 THE MAXIMUM PRINCIPLE 331
7.2.1 THE MAXIMUM PRINCIPLE 331
7.3 HIGHER ORDER EQUATIONS 334
7.3.1 "THE MEAN INEQUALITY" 334
7.3.2 EXISTENCE OF A CLASSICAL SOLUTION IN THE CASE AT = 2 337 7.3.3 THE
CASE TV = 3 339
7.3.4 CONTINUITY OF THE SOLUTION FOR THE LAPLACE OPERATOR ON I2 341
7.3.5 THE CASE OF THE RIGHT HAND SIDE EQUAL TO ZERO 344
NOTATION INDEX 365
SUBJECT INDEX 367
AUTHOR INDEX 371 |
| any_adam_object | 1 |
| author | Nečas, Jindřich 1929-2002 |
| author_GND | (DE-588)172282470 |
| author_facet | Nečas, Jindřich 1929-2002 |
| author_role | aut |
| author_sort | Nečas, Jindřich 1929-2002 |
| author_variant | j n jn |
| building | Verbundindex |
| bvnumber | BV036126140 |
| classification_rvk | SK 560 |
| ctrlnum | (OCoLC)759528567 (DE-599)DNB1000033643 |
| dewey-full | 515.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3533 |
| dewey-search | 515.3533 |
| dewey-sort | 3515.3533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV036126140 |
| illustrated | Illustrated |
| indexdate | 2024-09-06T00:16:50Z |
| institution | BVB |
| isbn | 9783642104541 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020208547 |
| oclc_num | 759528567 |
| open_access_boolean | |
| owner | DE-703 DE-384 DE-83 DE-19 DE-BY-UBM DE-188 DE-706 |
| owner_facet | DE-703 DE-384 DE-83 DE-19 DE-BY-UBM DE-188 DE-706 |
| physical | XVI, 372 S. graph. Darst. |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | Springer |
| record_format | marc |
| series2 | Springer monographs in mathematics |
| spelling | Nečas, Jindřich 1929-2002 Verfasser (DE-588)172282470 aut Les méthodes directes en théorie des équations elliptiques Direct methods in the theory of elliptic equations Jindřich Nečas Berlin [u.a.] Springer 2012 XVI, 372 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Lösung Chemie (DE-588)4036159-7 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Direkte Methode (DE-588)4705893-6 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Sobolev-Raum (DE-588)4055345-0 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 s Randwertproblem (DE-588)4048395-2 s Variationsrechnung (DE-588)4062355-5 s Direkte Methode (DE-588)4705893-6 s Sobolev-Raum (DE-588)4055345-0 s DE-604 Lösung Chemie (DE-588)4036159-7 s Erscheint auch als Online-Ausgabe 978-3-642-10455-8 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3423767&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208547&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nečas, Jindřich 1929-2002 Direct methods in the theory of elliptic equations Lösung Chemie (DE-588)4036159-7 gnd Randwertproblem (DE-588)4048395-2 gnd Direkte Methode (DE-588)4705893-6 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Sobolev-Raum (DE-588)4055345-0 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4036159-7 (DE-588)4048395-2 (DE-588)4705893-6 (DE-588)4014485-9 (DE-588)4055345-0 (DE-588)4062355-5 |
| title | Direct methods in the theory of elliptic equations |
| title_alt | Les méthodes directes en théorie des équations elliptiques |
| title_auth | Direct methods in the theory of elliptic equations |
| title_exact_search | Direct methods in the theory of elliptic equations |
| title_full | Direct methods in the theory of elliptic equations Jindřich Nečas |
| title_fullStr | Direct methods in the theory of elliptic equations Jindřich Nečas |
| title_full_unstemmed | Direct methods in the theory of elliptic equations Jindřich Nečas |
| title_short | Direct methods in the theory of elliptic equations |
| title_sort | direct methods in the theory of elliptic equations |
| topic | Lösung Chemie (DE-588)4036159-7 gnd Randwertproblem (DE-588)4048395-2 gnd Direkte Methode (DE-588)4705893-6 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Sobolev-Raum (DE-588)4055345-0 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Lösung Chemie Randwertproblem Direkte Methode Elliptische Differentialgleichung Sobolev-Raum Variationsrechnung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=3423767&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020208547&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT necasjindrich lesmethodesdirectesentheoriedesequationselliptiques AT necasjindrich directmethodsinthetheoryofellipticequations |