Unbounded functionals in the calculus of variations: representation, relaxation, and homogenization
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| Main Authors: | , |
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| Format: | Book |
| Language: | English |
| Published: |
Boca Raton [u.a.]
Chapman & Hall/CRC
2002
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| Series: | Chapman & Hall CRC monographs and surveys in pure and applied mathematics
125 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 394 S. |
| ISBN: | 1584882352 |
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Record in the Search Index
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| adam_text | K CHAPMAN & HALL/CRC MONOGRAPHS AND SURVEYS IN PURE AND APPLIED
MATHEMATICS 125 UNBOUNDED FUNCTIONALS IN THE CALCULUS OF VARIATIONS
REPRESENTATION, RELAXATION, AND HOMOGENIZATION LUCIANO CARBONE RICCARDO
DE ARCANGELIS CHAPMAN & HALL/CRC A CRC PRESS COMPANY BOCA RATON LONDON
NEW YORK WASHINGTON, D.C. TABLE OF CONTENTS PREFACE XI BASIC NOTATIONS
AND RECALLS 1 1 ELEMENTS OF CONVEX ANALYSIS 9 1.1 CONVEX SETS AND
FUNCTIONS 9 1.2 CONVEX AND LOWER SEMICONTINUOUS ENVELOPES IN R N 23 1.3
LOWER SEMICONTINUOUS ENVELOPES OF CONVEX ENVELOPES 29 1.4 CONVEX
ENVELOPES OF LOWER SEMICONTINUOUS ENVELOPES 35 2 ELEMENTS OF MEASURE AND
INCREASING SET FUNCTIONS THEORIES 45 2.1 MEASURES AND INTEGRALS 45 2.2
BASICS ON L P SPACES 55 2.3 DERIVATION OF MEASURES 60 2.4 ABSTRACT
MEASURE THEORY IN TOPOLOGICAL SETTINGS 63 2.5 LOCAL PROPERTIES OF
BOUNDARIES OF OPEN SUBSETS OF R. 67 2.6 INCREASING SET FUNCTIONS 70 2.7
INCREASING SET FUNCTIONAL 79 3 MINIMIZATION METHODS AND VARIATIONAL
CONVERGENCES 83 3.1 THE DIRECT METHODS IN THE CALCULUS OF VARIATIONS 83
3.2 T-CONVERGENCE 87 3.3 APPLICATIONS TO THE CALCULUS OF VARIATIONS 94
3.4 T-CONVERGENCE IN TOPOLOGICAL VECTOR SPACES AND OF INCREASING SET
FUNCTIONAL 99 -3.5 RELAXATION 102 4 BV AND SOBOLEV SPACES 107 4.1
REGULARIZATION OF MEASURES AND OF SUMMABLE FUNCTIONS 107 4.2 BV SPACES
113 4.3 SOBOLEV SPACES 121 VIII TABLE OF CONTENTS 4.4 SOME COMPACTNESS
CRITERIA 130 4.5 PERIODIC SOBOLEV FUNCTIONS 134 5 LOWER SEMICONTINUITY
AND MINIMIZATION OF INTEGRAL FUNCTIONAL 137 5.1 FUNCTIONAL ON BV SPACES
137 5.2 FUNCTIONAL ON SOBOLEV SPACES 142 5.3 MINIMIZATION OF INTEGRAL
FUNCTIONALS 144 6 CLASSICAL RESULTS AND MATHEMATICAL MODELS ORIGINATING
UNBOUNDED FUNCTIONALS 149 6.1 CLASSICAL UNIQUE EXTENSION RESULTS 149 6.2
CLASSICAL INTEGRAL REPRESENTATION RESULTS 150 6.3 CLASSICAL RELAXATION
RESULTS 153 6.4 CLASSICAL HOMOGENIZATION RESULTS 155 6.5 MATHEMATICAL
ASPECTS OF SOME PHYSICAL MODELS ORIGINATING UNBOUNDED FUNCTIONALS 157 7
ABSTRACT REGULARIZATION AND JENSEN S INEQUALITY 159 7.1 INTEGRAL OF
FUNCTIONS WITH VALUES IN LOCALLY CONVEX TOPOLOGICAL VECTOR SPACES 159
7.2 ON THE DEFINITION OF A FUNCTIONAL ON FUNCTIONS AND ON THEIR
EQUIVALENCE CLASSES 163 7.3 REGULARIZATION OF FUNCTIONS IN LOCALLY
CONVEX TOPOLOGICAL VECTOR SUBSPACES OF LJ OC {R N ) 165 7.4 APPLICATIONS
TO CONVEX FUNCTIONALS ON BV SPACES 169 8 UNIQUE EXTENSION RESULTS 177
8.1 UNIQUE EXTENSION RESULTS FOR INNER REGULAR FUNCTIONALS 178 8.2
EXISTENCE AND UNIQUENESS RESULTS 180 8.3 UNIQUE EXTENSION RESULTS FOR
MEASURE LIKE FUNCTIONALS 182 8.4 SOME APPLICATIONS 186 8.5 A NOTE ON
LAVRENTIEV PHENOMENON 189 9 INTEGRAL REPRESENTATION FOR UNBOUNDED
FUNCTIONALS 191 9.1 REPRESENTATION ON LINEAR FUNCTIONS 191 9.2
REPRESENTATION ON CONTINUOUSLY DIFFERENTIABLE FUNCTIONS 192 9.3
REPRESENTATION ON SOBOLEV SPACES 199 9.4 REPRESENTATION ON BV SPACES 207
10 RELAXATION OF UNBOUNDED FUNCTIONALS 211 10.1 NOTATIONS AND ELEMENTARY
PROPERTIES OF RELAXED FUNCTIONALS IN THE NEUMANN CASE 211 10.2
RELAXATION OF NEUMANN PROBLEMS: THE CASE OF BOUNDED EFFECTIVE DOMAIN
WITH NONEMPTY INTERIOR 214 10.3 RELAXATION OF NEUMANN PROBLEMS: THE CASE
OF BOUNDED TABLE OF CONTENTS IX EFFECTIVE DOMAIN WITH EMPTY INTERIOR 219
10.4 RELAXATION OF NEUMANN PROBLEMS: A FIRST RESULT WITHOUT BOUNDEDNESS
ASSUMPTIONS ON THE EFFECTIVE DOMAIN 226 10.5 RELAXATION OF NEUMANN
PROBLEMS: RELAXATION IN BV SPACES 229 10.6 NOTATIONS AND ELEMENTARY
PROPERTIES OF RELAXED FUNCTIONALS IN THE DIRICHLET CASE 232 10.7
RELAXATION OF DIRICHLET PROBLEMS 234 10.8 APPLICATIONS TO MINIMUM
PROBLEMS 246 10.9 ADDITIONAL REMARKS ON INTEGRAL REPRESENTATION ON THE
WHOLE SPACE OF LIPSCHITZ FUNCTIONS 253 11 CUT-OFF FUNCTIONS AND
PARTITIONS OF UNITY 261 11.1 CUT-OFF FUNCTIONS 261 11.2 PARTITIONS OF
UNITY 269 12 HOMOGENIZATION OF UNBOUNDED FUNCTIONALS 273 12.1 NOTATIONS
AND BASIC RESULTS 274 12.2 SOME PROPERTIES OF T-LIMITS 280 12.3
FINITENESS CONDITIONS 287 12.4 REPRESENTATION ON LINEAR FUNCTIONS 293
12.5 A BLOW-UP CONDITION 306 12.6 REPRESENTATION RESULTS 307 12.7
APPLICATIONS TO THE CONVERGENCE OF MINIMA AND OF MINIMIZERS 311 13
HOMOGENIZATION OF UNBOUNDED FUNCTIONALS WITH SPECIAL CONSTRAINTS 319
13.1 HOMOGENIZATION WITH FIXED CONSTRAINTS: THE CASE OF NEUMANN BOUNDARY
CONDITIONS 319 13.2 HOMOGENIZATION WITH FIXED CONSTRAINTS: THE CASE OF
DIRICHLET BOUNDARY CONDITIONS 328 13.3 HOMOGENIZATION WITH FIXED
CONSTRAINTS: THE CASE OF MIXED BOUNDARY CONDITIONS 331 13.4
HOMOGENIZATION WITH FIXED CONSTRAINTS: APPLICATIONS TO THE CONVERGENCE
OF MINIMA AND OF MINIMIZERS 335 13.5 HOMOGENIZATION WITH OSCILLATING
SPECIAL CONSTRAINTS 345 13.6 FINAL REMARKS 351 14 SOME EXPLICIT
COMPUTATIONS OF HOMOGENIZED ENERGIES IN MATHEMATICAL MODELS ORIGINATING
UNBOUNDED FUNCTIONALS 353 14.1 HOMOGENIZATION IN ELASTIC-PLASTIC TORSION
353 14.2 HOMOGENIZATION IN THE MODELLING OF NONLINEAR ELASTOMERS... 359
14.3 HOMOGENIZATION IN ELECTROSTATIC SCREENING 364 X TABLE OF CONTENTS
BIBLIOGRAPHY 375 LIST OF SYMBOLS 389 INDEX 391
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| physical | XIII, 394 S. |
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| series2 | Chapman & Hall CRC monographs and surveys in pure and applied mathematics |
| spelling | Carbone, Luciano Verfasser aut Unbounded functionals in the calculus of variations representation, relaxation, and homogenization Luciano Carbone ; Riccardo De Arcangelis Boca Raton [u.a.] Chapman & Hall/CRC 2002 XIII, 394 S. txt rdacontent n rdamedia nc rdacarrier Chapman & Hall CRC monographs and surveys in pure and applied mathematics 125 Calculus of variations Functionals Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Funktional (DE-588)4155667-7 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 s Funktional (DE-588)4155667-7 s DE-604 Arcangelis, Riccardo de Verfasser aut Chapman & Hall CRC monographs and surveys in pure and applied mathematics 125 (DE-604)BV013350872 125 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009878116&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Carbone, Luciano Arcangelis, Riccardo de Unbounded functionals in the calculus of variations representation, relaxation, and homogenization Chapman & Hall CRC monographs and surveys in pure and applied mathematics Calculus of variations Functionals Variationsrechnung (DE-588)4062355-5 gnd Funktional (DE-588)4155667-7 gnd |
| subject_GND | (DE-588)4062355-5 (DE-588)4155667-7 |
| title | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization |
| title_auth | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization |
| title_exact_search | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization |
| title_full | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization Luciano Carbone ; Riccardo De Arcangelis |
| title_fullStr | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization Luciano Carbone ; Riccardo De Arcangelis |
| title_full_unstemmed | Unbounded functionals in the calculus of variations representation, relaxation, and homogenization Luciano Carbone ; Riccardo De Arcangelis |
| title_short | Unbounded functionals in the calculus of variations |
| title_sort | unbounded functionals in the calculus of variations representation relaxation and homogenization |
| title_sub | representation, relaxation, and homogenization |
| topic | Calculus of variations Functionals Variationsrechnung (DE-588)4062355-5 gnd Funktional (DE-588)4155667-7 gnd |
| topic_facet | Calculus of variations Functionals Variationsrechnung Funktional |
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