Homogenization of partial differential equations:
Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to...
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English Russian |
| Veröffentlicht: |
Boston u.a.
Birkhäuser
2006
|
| Schriftenreihe: | Progress in mathematical physics
46 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems describing various physical processes in micro-inhomogeneous media. From the technical viewpoint the work focuses on the construction of nonstandard models for media characterized by several small-scale parameters (multiscale models). A variety of techniques are used -- specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. This new method can be applied to a wide variety of problems. Key topics in this systematic exposition include asymptotic analysis, Dirichlet- and Neumann-type boundary value problems, differential equations with rapidly oscillating coefficients, homogenization, homogenized and non-local models. Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Applied mathematicians, advanced-level graduate students, physicists, engineers, and specialists in mechanics will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text TOC:Introduction - The Dirichlet boundary value problem in strongly perforated domains - The Dirichlet boundary value problem in domains with complex boundary - Strongly connected domains - The Neumann boundary value problems in strongly connected domains - Non-stationary problems and spectral problems - Differential equations with rapidly oscillating coefficients - Homogenized conjugation conditions - Bibliograph |
| Beschreibung: | Aus d. Russ. übers. |
| Beschreibung: | XII, 398 S. |
| ISBN: | 0817643516 |
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| 245 | 1 | 0 | |a Homogenization of partial differential equations |c Vladimir A. Marchenko ; Evgueni Ya. Khruslov |
| 264 | 1 | |a Boston u.a. |b Birkhäuser |c 2006 | |
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| 490 | 1 | |a Progress in mathematical physics |v 46 | |
| 500 | |a Aus d. Russ. übers. | ||
| 520 | |a Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems describing various physical processes in micro-inhomogeneous media. From the technical viewpoint the work focuses on the construction of nonstandard models for media characterized by several small-scale parameters (multiscale models). A variety of techniques are used -- | ||
| 520 | |a specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. This new method can be applied to a wide variety of problems. Key topics in this systematic exposition include asymptotic analysis, Dirichlet- and Neumann-type boundary value problems, differential equations with rapidly oscillating coefficients, homogenization, homogenized and non-local models. Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Applied mathematicians, advanced-level graduate students, physicists, engineers, and specialists in mechanics will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text | ||
| 520 | |a TOC:Introduction - The Dirichlet boundary value problem in strongly perforated domains - The Dirichlet boundary value problem in domains with complex boundary - Strongly connected domains - The Neumann boundary value problems in strongly connected domains - Non-stationary problems and spectral problems - Differential equations with rapidly oscillating coefficients - Homogenized conjugation conditions - Bibliograph | ||
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Datensatz im Suchindex
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| adam_text | VLADIMIR A. MARCHENKO EVGUENI YA. KHRUSLOV HOMOGENIZATION OF PARTIAL
DIFFERENTIAL EQUATIONS TRANSLATED FROM THE ORIGINAL RUSSIAN BY M.
GONCHARENKO AND D. SHEPELSKY BIRKHAUSER BOSTON * BASEL * BERLIN CONTENTS
PREFACE V 1 INTRODUCTION 1 1.1 THE SIMPLEST HOMOGENIZED MODEL 1 1.2
NONLOCAL HOMOGENIZED MODEL 5 1.3 TWO-COMPONENT HOMOGENIZED MODEL 7 1.4
HOMOGENIZED MODEL WITH MEMORY 10 1.5 HOMOGENIZED MODEL WITH MEMORY: THE
CASE OF VIOLATED UNIFORM BOUNDEDNESS OF B (X) 12 1.6 HOMOGENIZATION OF
BOUNDARY VALUE PROBLEMS IN STRONGLY PERFORATED DOMAINS 13 1.7 STRONGLY
CONNECTED AND WEAKLY CONNECTED DOMAINS: DEFINITIONS AND QUANTITATIVE
CHARACTERISTICS 18 1.7.1 STRONGLY CONNECTED AND WEAKLY CONNECTED DOMAINS
18 1.7.2 LOCAL MESOSCOPIC CHARACTERISTICS OF STRONGLY CONNECTED DOMAINS
20 2 THE DIRICHLET BOUNDARY VALUE PROBLEM IN STRONGLY PERFORATED DOMAINS
WITH FINE-GRAINED BOUNDARY 31 2.1 METHOD OF ORTHOGONAL PROJECTIONS AND
AN ABSTRACT SCHEME FOR THE DIRICHLET PROBLEM IN STRONGLY DERFORATED
DOMAINS 32 2.1.1 METHOD OF ORTHOGONAL PROJECTIONS 32 2.1.2 AN ABSTRACT
SCHEME 34 2.2 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE DIRICHLET PROBLEM
IN DOMAINS WITH FINE-GRAINED BOUNDARY 42 2.2.1 PROBLEM FORMULATION AND
MAIN RESULT 42 2.2.2 AUXILIARY STATEMENTS 44 2.2.3 VALIDITY OF THE
ASSUMPTIONS OF THEOREM 2.3 49 2.2.4 PROOF OF THEOREM 2.3 55 2.3 THE
DIRICHLET PROBLEM IN DOMAINS WITH RANDOM FINE-GRAINED BOUNDARY 56
CONTENTS 2.3.1 PROBLEM FORMULATION AND MAIN RESULT 56 2.3.2 ASSUMPTIONS
OF THEOREM 2.3 IN PROBABILITY 58 2.3.3 CONVERGENCE IN PROBABILITY OF
SOLUTIONS OF PROBLEM (2.23M2.24) 63 THE DIRICHLET BOUNDARY VALUE PROBLEM
IN STRONGLY PERFORATED DOMAINS WITH COMPLEX BOUNDARY 67 3.1 NECESSARY
AND SUFFICIENT CONDITIONS FOR CONVERGENCE OF SOLUTIONS OF THE DIRICHLET
PROBLEM 67 3.1.1 PROBLEM FORMULATION AND MAIN RESULT 67 3.1.2
SUFFICIENCY OF CONDITIONS 1 AND 2 68 3.1.3 NECESSITY OF CONDITIONS 1 AND
2 77 3.1.4 HIGHER-ORDER EQUATIONS 80 3.2 ASYMPTOTIC BEHAVIOR OF
SOLUTIONS OF VARIATIONAL PROBLEMS FOR NONQUADRATIC FUNCTIONALS IN
DOMAINS WITH COMPLEX BOUNDARY .... 82 3.2.1 THE SOBOLEV-ORLICZ SPACES:
PRELIMINARIES 82 3.2.2 PROBLEM STATEMENT AND MAIN RESULT 84 3.2.3 THE
PROOF OF THEOREM 3.7 86 3.3 ASYMPTOTIC BEHAVIOR OF THE POTENTIAL OF THE
ELECTROSTATIC FIELD IN A WEAKLY NONLINEAR MEDIUM WITH THIN PERFECTLY
CONDUCTING FILAMENTS 96 STRONGLY CONNECTED DOMAINS 105 4.1 PRELIMINARY
CONSIDERATIONS 106 4.1.1 ON ONE PROPERTY OF LATTICES WITH COLORED NODES
106 4.1.2 SOME PROPERTIES OF DIFFERENTIABLE FUNCTIONS 107 4.2 STRONGLY
CONNECTED DOMAINS 114 4.2.1 CONVERGENCE AND COMPACTNESS OF SEQUENCES OF
FUNCTIONS GIVEN IN VARYING DOMAINS 114 4.2.2 DOMAINS ADMITTING EXTENSION
OF FUNCTIONS 116 4.2.3 DOMAINS ADMITTING EXTENSION OF FUNCTIONS WITH
SMALL DISTORTION 121 4.3 STRONGLY CONNECTED DOMAINS OF DECREASING VOLUME
125 4.3.1 CONVERGENCE AND COMPACTNESS OF SEQUENCES OF FUNCTIONS DEFINED
IN DOMAINS OF DECREASING VOLUME 125 4.3.2 EXAMPLES OF DOMAINS OF
DECREASING VOLUME THAT SATISFY THE STRONG CONNECTIVITY CONDITION A 128
THE NEUMANN BOUNDARY VALUE PROBLEMS IN STRONGLY PERFORATED DOMAINS 137
5.1 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY VALUE PROBLEMS IN
STRONGLY CONNECTED DOMAINS 137 5.1.1 THE CONDUCTIVITY TENSOR: MAIN
THEOREM 137 5.1.2 PROOF OF THEOREM 5.1 139 5.1.3 CONVERGENCE OF ENERGIES
AND FLOWS 145 CONTENTS XI 5.1.4 NECESSITY OF CONDITIONS 1 AND 2 OF
THEOREM 5.1 147 5.2 CALCULATION OF THE CONDUCTIVITY TENSOR FOR
STRUCTURES CLOSE TO PERIODIC 151 5.3 ASYMPTOTIC BEHAVIOR OF THE NEUMANN
BOUNDARY VALUE PROBLEMS IN WEAKLY CONNECTED DOMAINS 158 5.3.1 WEAKLY
CONNECTED DOMAINS 158 5.3.2 QUANTITATIVE CHARACTERISTICS OF WEAKLY
CONNECTED DOMAINS: MAIN THEOREM 160 5.3.3 AUXILIARY CONSTRUCTIONS AND
STATEMENTS 163 5.3.4 PROOF OF THEOREM 5.7 173 5.3.5 CONVERGENCE OF
ENERGIES AND FLOWS 178 5.4 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY
VALUE PROBLEMS IN DOMAINS WITH ACCUMULATORS (TRAPS) 180 5.4.1 WEAKLY
CONNECTED DOMAINS WITH ACCUMULATORS AND THEIR QUANTITATIVE
CHARACTERISTICS: MAIN THEOREM 180 5.4.2 AUXILIARY CONSTRUCTIONS AND
STATEMENTS 183 5.4.3 PROOF OF THEOREM 5.13 189 5.4.4 A GENERALIZATION OF
THEOREM 5.13 196 5.5 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY VALUE
PROBLEMS IN STRONGLY CONNECTED DOMAINS OF DECREASING VOLUME 198 5.5.1
QUANTITATIVE CHARACTERISTICS OF DOMAINS AND MAIN THEOREM 198 5.5.2
EXAMPLES 202 NONSTATIONARY PROBLEMS AND SPECTRAL PROBLEMS 211 6.1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONSTATIONARY PROBLEMS IN TUBE
DOMAINS 211 6.1.1 CONVERGENCE OF SPECTRAL PROJECTIONS 211 6.1.2 THE
DIRICHLET INITIAL BOUNDARY VALUE PROBLEM 213 6.1.3 THE NEUMANN INITIAL
BOUNDARY VALUE PROBLEM 216 6.1.4 THE NEUMANN INITIAL BOUNDARY VALUE
PROBLEM IN DOMAINS WITH ACCUMULATORS 218 6.2 ASYMPTOTIC BEHAVIOR OF
SOLUTIONS OF DIRICHLET PROBLEMS IN VARYING STRONGLY PERFORATED DOMAINS
220 6.2.1 PROBLEM FORMULATION AND MAIN RESULT 220 6.2.2 ESTIMATES FOR
DERIVATIVES OF W (S) (X,T) 222 6.2.3 PROOF OF THEOREM 6.7 226 6.3
ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF BOUNDARY VALUE PROBLEMS IN
STRONGLY PERFORATED DOMAINS 228 6.3.1 STRONGLY CONNECTED DOMAINS 228
6.3.2 WEAKLY CONNECTED DOMAINS 233 6.3.3 DOMAINS WITH ACCUMULATORS 234
DIFFERENTIAL EQUATIONS WITH RAPIDLY OSCILLATING COEFFICIENTS 23 7 7.1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH
COEFFICIENTS THAT ARE NOT UNIFORMLY ELLIPTIC 239 XII CONTENTS 7.1.1
PROBLEM FORMULATION AND MAIN THEOREM 239 7.1.2 AUXILIARY STATEMENTS AND
CONSTRUCTIONS 244 7.1.3 A STATIONARY VERSION OF THEOREM 7.1 262 7.1.4
COMPLETION OF THE PROOF OF THEOREM 7.1 268 7.2 EXAMPLES OF PARTICULAR
REALIZATIONS OF THE HOMOGENIZED DIFFUSION MODEL 272 7.2.1 ONE-PHASE
MODEL WITH MEMORY 272 7.2.2 HOMOGENIZED DIFFUSION MODEL FOR MEDIA WITH
TRAPS 276 7.2.3 TWO-COMPONENT MODELS 281 7.2.4 A PROBABILISTIC PROBLEM
283 7.3 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH
COEFFICIENTS THAT ARE NOT UNIFORMLY BOUNDED 287 7.3.1 STATIONARY
PROBLEM: MAIN THEOREM 287 7.3.2 AUXILIARY STATEMENTS 291 7.3.3 PROOF OF
THEOREM 7.13 302 7.3.4 NONSTATIONARY PROBLEMS 307 7.4 AN EXAMPLE OF A
NONLOCAL HOMOGENIZED MODEL 309 7.5 HOMOGENIZED HEAT CONDUCTION MODEL FOR
A MEDIUM CONTAINING INCLUSIONS WITH HIGH HEAT CAPACITY 312 7.5.1 PROBLEM
STATEMENT AND MAIN RESULT 312 7.5.2 A STATIONARY PROBLEM 314 7.5.3 PROOF
OF THEOREM 7.21 322 7.5.4 FINE-GRAINED PERIODIC INCLUSIONS 327 8
HOMOGENIZED CONJUGATION CONDITIONS 333 8.1 THE DIRICHLET PROBLEM:
SURFACE DISTRIBUTION OF SETS F (S) 333 8.1.1 PROBLEM FORMULATION AND
MAIN RESULT 333 8.1.2 A PREPARATORY LEMMA 336 8.1.3 MAIN PART OF THE
PROOF OF THEOREM 8.1 345 8.2 THE NEUMANN PROBLEM: SURFACE DISTRIBUTION
OF INCLUSIONS 350 8.2.1 PROBLEM FORMULATION AND MAIN RESULT 350 8.2.2 A
PREPARATORY LEMMA 353 8.2.3 MAIN PART OF THE PROOF OF THEOREM 8.7 357
8.3 DEFLECTION OF ELASTIC PLATES 361 8.3.1 RIGIDLY FIXED PLATES 362
8.3.2 FREE PLATE 370 8.4 HOMOGENIZED CONJUGATION CONDITIONS FOR THE
GINZBURG-LANDAU EQUATION; STATIONARY JOSEPHSON EFFECT 377 8.4.1 WEAKLY
CONNECTED CONDUCTORS 377 8.4.2 CONVERGENCE THEOREM 380 8.4.3 STATIONARY
JOSEPHSON EFFECT 384 REFERENCES 387 INDEX 397
|
| adam_txt |
VLADIMIR A. MARCHENKO EVGUENI YA. KHRUSLOV HOMOGENIZATION OF PARTIAL
DIFFERENTIAL EQUATIONS TRANSLATED FROM THE ORIGINAL RUSSIAN BY M.
GONCHARENKO AND D. SHEPELSKY BIRKHAUSER BOSTON * BASEL * BERLIN CONTENTS
PREFACE V 1 INTRODUCTION 1 1.1 THE SIMPLEST HOMOGENIZED MODEL 1 1.2
NONLOCAL HOMOGENIZED MODEL 5 1.3 TWO-COMPONENT HOMOGENIZED MODEL 7 1.4
HOMOGENIZED MODEL WITH MEMORY 10 1.5 HOMOGENIZED MODEL WITH MEMORY: THE
CASE OF VIOLATED UNIFORM BOUNDEDNESS OF B (X) 12 1.6 HOMOGENIZATION OF
BOUNDARY VALUE PROBLEMS IN STRONGLY PERFORATED DOMAINS 13 1.7 STRONGLY
CONNECTED AND WEAKLY CONNECTED DOMAINS: DEFINITIONS AND QUANTITATIVE
CHARACTERISTICS 18 1.7.1 STRONGLY CONNECTED AND WEAKLY CONNECTED DOMAINS
18 1.7.2 LOCAL MESOSCOPIC CHARACTERISTICS OF STRONGLY CONNECTED DOMAINS
20 2 THE DIRICHLET BOUNDARY VALUE PROBLEM IN STRONGLY PERFORATED DOMAINS
WITH FINE-GRAINED BOUNDARY 31 2.1 METHOD OF ORTHOGONAL PROJECTIONS AND
AN ABSTRACT SCHEME FOR THE DIRICHLET PROBLEM IN STRONGLY DERFORATED
DOMAINS 32 2.1.1 METHOD OF ORTHOGONAL PROJECTIONS 32 2.1.2 AN ABSTRACT
SCHEME 34 2.2 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF THE DIRICHLET PROBLEM
IN DOMAINS WITH FINE-GRAINED BOUNDARY 42 2.2.1 PROBLEM FORMULATION AND
MAIN RESULT 42 2.2.2 AUXILIARY STATEMENTS 44 2.2.3 VALIDITY OF THE
ASSUMPTIONS OF THEOREM 2.3 49 2.2.4 PROOF OF THEOREM 2.3 55 2.3 THE
DIRICHLET PROBLEM IN DOMAINS WITH RANDOM FINE-GRAINED BOUNDARY 56
CONTENTS 2.3.1 PROBLEM FORMULATION AND MAIN RESULT 56 2.3.2 ASSUMPTIONS
OF THEOREM 2.3 "IN PROBABILITY" 58 2.3.3 CONVERGENCE IN PROBABILITY OF
SOLUTIONS OF PROBLEM (2.23M2.24) 63 THE DIRICHLET BOUNDARY VALUE PROBLEM
IN STRONGLY PERFORATED DOMAINS WITH COMPLEX BOUNDARY 67 3.1 NECESSARY
AND SUFFICIENT CONDITIONS FOR CONVERGENCE OF SOLUTIONS OF THE DIRICHLET
PROBLEM 67 3.1.1 PROBLEM FORMULATION AND MAIN RESULT 67 3.1.2
SUFFICIENCY OF CONDITIONS 1 AND 2 68 3.1.3 NECESSITY OF CONDITIONS 1 AND
2 77 3.1.4 HIGHER-ORDER EQUATIONS 80 3.2 ASYMPTOTIC BEHAVIOR OF
SOLUTIONS OF VARIATIONAL PROBLEMS FOR NONQUADRATIC FUNCTIONALS IN
DOMAINS WITH COMPLEX BOUNDARY . 82 3.2.1 THE SOBOLEV-ORLICZ SPACES:
PRELIMINARIES 82 3.2.2 PROBLEM STATEMENT AND MAIN RESULT 84 3.2.3 THE
PROOF OF THEOREM 3.7 86 3.3 ASYMPTOTIC BEHAVIOR OF THE POTENTIAL OF THE
ELECTROSTATIC FIELD IN A WEAKLY NONLINEAR MEDIUM WITH THIN PERFECTLY
CONDUCTING FILAMENTS 96 STRONGLY CONNECTED DOMAINS 105 4.1 PRELIMINARY
CONSIDERATIONS 106 4.1.1 ON ONE PROPERTY OF LATTICES WITH COLORED NODES
106 4.1.2 SOME PROPERTIES OF DIFFERENTIABLE FUNCTIONS 107 4.2 STRONGLY
CONNECTED DOMAINS 114 4.2.1 CONVERGENCE AND COMPACTNESS OF SEQUENCES OF
FUNCTIONS GIVEN IN VARYING DOMAINS 114 4.2.2 DOMAINS ADMITTING EXTENSION
OF FUNCTIONS 116 4.2.3 DOMAINS ADMITTING EXTENSION OF FUNCTIONS WITH
SMALL DISTORTION 121 4.3 STRONGLY CONNECTED DOMAINS OF DECREASING VOLUME
125 4.3.1 CONVERGENCE AND COMPACTNESS OF SEQUENCES OF FUNCTIONS DEFINED
IN DOMAINS OF DECREASING VOLUME 125 4.3.2 EXAMPLES OF DOMAINS OF
DECREASING VOLUME THAT SATISFY THE STRONG CONNECTIVITY CONDITION A 128
THE NEUMANN BOUNDARY VALUE PROBLEMS IN STRONGLY PERFORATED DOMAINS 137
5.1 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY VALUE PROBLEMS IN
STRONGLY CONNECTED DOMAINS 137 5.1.1 THE CONDUCTIVITY TENSOR: MAIN
THEOREM 137 5.1.2 PROOF OF THEOREM 5.1 139 5.1.3 CONVERGENCE OF ENERGIES
AND FLOWS 145 CONTENTS XI 5.1.4 NECESSITY OF CONDITIONS 1 AND 2 OF
THEOREM 5.1 147 5.2 CALCULATION OF THE CONDUCTIVITY TENSOR FOR
STRUCTURES CLOSE TO PERIODIC 151 5.3 ASYMPTOTIC BEHAVIOR OF THE NEUMANN
BOUNDARY VALUE PROBLEMS IN WEAKLY CONNECTED DOMAINS 158 5.3.1 WEAKLY
CONNECTED DOMAINS 158 5.3.2 QUANTITATIVE CHARACTERISTICS OF WEAKLY
CONNECTED DOMAINS: MAIN THEOREM 160 5.3.3 AUXILIARY CONSTRUCTIONS AND
STATEMENTS 163 5.3.4 PROOF OF THEOREM 5.7 173 5.3.5 CONVERGENCE OF
ENERGIES AND FLOWS 178 5.4 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY
VALUE PROBLEMS IN DOMAINS WITH ACCUMULATORS (TRAPS) 180 5.4.1 WEAKLY
CONNECTED DOMAINS WITH ACCUMULATORS AND THEIR QUANTITATIVE
CHARACTERISTICS: MAIN THEOREM 180 5.4.2 AUXILIARY CONSTRUCTIONS AND
STATEMENTS 183 5.4.3 PROOF OF THEOREM 5.13 189 5.4.4 A GENERALIZATION OF
THEOREM 5.13 196 5.5 ASYMPTOTIC BEHAVIOR OF THE NEUMANN BOUNDARY VALUE
PROBLEMS IN STRONGLY CONNECTED DOMAINS OF DECREASING VOLUME 198 5.5.1
QUANTITATIVE CHARACTERISTICS OF DOMAINS AND MAIN THEOREM 198 5.5.2
EXAMPLES 202 NONSTATIONARY PROBLEMS AND SPECTRAL PROBLEMS 211 6.1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF NONSTATIONARY PROBLEMS IN TUBE
DOMAINS 211 6.1.1 CONVERGENCE OF SPECTRAL PROJECTIONS 211 6.1.2 THE
DIRICHLET INITIAL BOUNDARY VALUE PROBLEM 213 6.1.3 THE NEUMANN INITIAL
BOUNDARY VALUE PROBLEM 216 6.1.4 THE NEUMANN INITIAL BOUNDARY VALUE
PROBLEM IN DOMAINS WITH ACCUMULATORS 218 6.2 ASYMPTOTIC BEHAVIOR OF
SOLUTIONS OF DIRICHLET PROBLEMS IN VARYING STRONGLY PERFORATED DOMAINS
220 6.2.1 PROBLEM FORMULATION AND MAIN RESULT 220 6.2.2 ESTIMATES FOR
DERIVATIVES OF W (S) (X,T) 222 6.2.3 PROOF OF THEOREM 6.7 226 6.3
ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF BOUNDARY VALUE PROBLEMS IN
STRONGLY PERFORATED DOMAINS 228 6.3.1 STRONGLY CONNECTED DOMAINS 228
6.3.2 WEAKLY CONNECTED DOMAINS 233 6.3.3 DOMAINS WITH ACCUMULATORS 234
DIFFERENTIAL EQUATIONS WITH RAPIDLY OSCILLATING COEFFICIENTS 23 7 7.1
ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH
COEFFICIENTS THAT ARE NOT UNIFORMLY ELLIPTIC 239 XII CONTENTS 7.1.1
PROBLEM FORMULATION AND MAIN THEOREM 239 7.1.2 AUXILIARY STATEMENTS AND
CONSTRUCTIONS 244 7.1.3 A STATIONARY VERSION OF THEOREM 7.1 262 7.1.4
COMPLETION OF THE PROOF OF THEOREM 7.1 268 7.2 EXAMPLES OF PARTICULAR
REALIZATIONS OF THE HOMOGENIZED DIFFUSION MODEL 272 7.2.1 ONE-PHASE
MODEL WITH MEMORY 272 7.2.2 HOMOGENIZED DIFFUSION MODEL FOR MEDIA WITH
TRAPS 276 7.2.3 TWO-COMPONENT MODELS 281 7.2.4 A PROBABILISTIC PROBLEM
283 7.3 ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF DIFFERENTIAL EQUATIONS WITH
COEFFICIENTS THAT ARE NOT UNIFORMLY BOUNDED 287 7.3.1 STATIONARY
PROBLEM: MAIN THEOREM 287 7.3.2 AUXILIARY STATEMENTS 291 7.3.3 PROOF OF
THEOREM 7.13 302 7.3.4 NONSTATIONARY PROBLEMS 307 7.4 AN EXAMPLE OF A
NONLOCAL HOMOGENIZED MODEL 309 7.5 HOMOGENIZED HEAT CONDUCTION MODEL FOR
A MEDIUM CONTAINING INCLUSIONS WITH HIGH HEAT CAPACITY 312 7.5.1 PROBLEM
STATEMENT AND MAIN RESULT 312 7.5.2 A STATIONARY PROBLEM 314 7.5.3 PROOF
OF THEOREM 7.21 322 7.5.4 FINE-GRAINED PERIODIC INCLUSIONS 327 8
HOMOGENIZED CONJUGATION CONDITIONS 333 8.1 THE DIRICHLET PROBLEM:
SURFACE DISTRIBUTION OF SETS F (S) 333 8.1.1 PROBLEM FORMULATION AND
MAIN RESULT 333 8.1.2 A PREPARATORY LEMMA 336 8.1.3 MAIN PART OF THE
PROOF OF THEOREM 8.1 345 8.2 THE NEUMANN PROBLEM: SURFACE DISTRIBUTION
OF INCLUSIONS 350 8.2.1 PROBLEM FORMULATION AND MAIN RESULT 350 8.2.2 A
PREPARATORY LEMMA 353 8.2.3 MAIN PART OF THE PROOF OF THEOREM 8.7 357
8.3 DEFLECTION OF ELASTIC PLATES 361 8.3.1 RIGIDLY FIXED PLATES 362
8.3.2 FREE PLATE 370 8.4 HOMOGENIZED CONJUGATION CONDITIONS FOR THE
GINZBURG-LANDAU EQUATION; STATIONARY JOSEPHSON EFFECT 377 8.4.1 WEAKLY
CONNECTED CONDUCTORS 377 8.4.2 CONVERGENCE THEOREM 380 8.4.3 STATIONARY
JOSEPHSON EFFECT 384 REFERENCES 387 INDEX 397 |
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| any_adam_object_boolean | 1 |
| author | Marčenko, Vladimir A. Chruslov, Evgenij J. |
| author_facet | Marčenko, Vladimir A. Chruslov, Evgenij J. |
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| author_sort | Marčenko, Vladimir A. |
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| bvnumber | BV021277052 |
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| ctrlnum | (OCoLC)181436886 (DE-599)BVBBV021277052 |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV021277052 |
| illustrated | Not Illustrated |
| index_date | 2024-07-02T13:46:11Z |
| indexdate | 2024-07-09T20:34:30Z |
| institution | BVB |
| isbn | 0817643516 |
| language | English Russian |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-014598090 |
| oclc_num | 181436886 |
| open_access_boolean | |
| owner | DE-384 DE-824 DE-83 DE-11 DE-188 |
| owner_facet | DE-384 DE-824 DE-83 DE-11 DE-188 |
| physical | XII, 398 S. |
| publishDate | 2006 |
| publishDateSearch | 2006 |
| publishDateSort | 2006 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematical physics |
| series2 | Progress in mathematical physics |
| spelling | Marčenko, Vladimir A. Verfasser aut Usredennye modeli mikroneodnorodnych sred Homogenization of partial differential equations Vladimir A. Marchenko ; Evgueni Ya. Khruslov Boston u.a. Birkhäuser 2006 XII, 398 S. txt rdacontent n rdamedia nc rdacarrier Progress in mathematical physics 46 Aus d. Russ. übers. Homogenization is a method for modelling processes in complex structures. These processes are far too complex for analytic and numerical methods and are best described by PDEs with rapidly oscillating coefficients - a technique that has become increasingly important in the last three decades due to its multiple applications in the areas of optimization, radiophysics, filtration theory, rheology, elasticity theory, and other domains of mechanics, physics, and technology. The present monograph is a comprehensive study of homogenization problems describing various physical processes in micro-inhomogeneous media. From the technical viewpoint the work focuses on the construction of nonstandard models for media characterized by several small-scale parameters (multiscale models). A variety of techniques are used -- specifically functional analysis, the spectral theory for differential operators, the Laplace transform, and, most importantly, a new variational PDE method for studying the asymptotic behavior of solutions of stationary boundary value problems. This new method can be applied to a wide variety of problems. Key topics in this systematic exposition include asymptotic analysis, Dirichlet- and Neumann-type boundary value problems, differential equations with rapidly oscillating coefficients, homogenization, homogenized and non-local models. Along with complete proofs of all main results, numerous examples of typical structures of micro-inhomogeneous media with their corresponding homogenized models are provided. Applied mathematicians, advanced-level graduate students, physicists, engineers, and specialists in mechanics will benefit from this monograph, which may be used in the classroom or as a comprehensive reference text TOC:Introduction - The Dirichlet boundary value problem in strongly perforated domains - The Dirichlet boundary value problem in domains with complex boundary - Strongly connected domains - The Neumann boundary value problems in strongly connected domains - Non-stationary problems and spectral problems - Differential equations with rapidly oscillating coefficients - Homogenized conjugation conditions - Bibliograph Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Homogenisierungsmethode (DE-588)4257770-6 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Homogenisierungsmethode (DE-588)4257770-6 s DE-604 Chruslov, Evgenij J. Verfasser aut Progress in mathematical physics 46 (DE-604)BV013823265 46 GBV Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014598090&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Marčenko, Vladimir A. Chruslov, Evgenij J. Homogenization of partial differential equations Progress in mathematical physics Partielle Differentialgleichung (DE-588)4044779-0 gnd Homogenisierungsmethode (DE-588)4257770-6 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4257770-6 |
| title | Homogenization of partial differential equations |
| title_alt | Usredennye modeli mikroneodnorodnych sred |
| title_auth | Homogenization of partial differential equations |
| title_exact_search | Homogenization of partial differential equations |
| title_exact_search_txtP | Homogenization of partial differential equations |
| title_full | Homogenization of partial differential equations Vladimir A. Marchenko ; Evgueni Ya. Khruslov |
| title_fullStr | Homogenization of partial differential equations Vladimir A. Marchenko ; Evgueni Ya. Khruslov |
| title_full_unstemmed | Homogenization of partial differential equations Vladimir A. Marchenko ; Evgueni Ya. Khruslov |
| title_short | Homogenization of partial differential equations |
| title_sort | homogenization of partial differential equations |
| topic | Partielle Differentialgleichung (DE-588)4044779-0 gnd Homogenisierungsmethode (DE-588)4257770-6 gnd |
| topic_facet | Partielle Differentialgleichung Homogenisierungsmethode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014598090&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV013823265 |
| work_keys_str_mv | AT marcenkovladimira usredennyemodelimikroneodnorodnychsred AT chruslovevgenijj usredennyemodelimikroneodnorodnychsred AT marcenkovladimira homogenizationofpartialdifferentialequations AT chruslovevgenijj homogenizationofpartialdifferentialequations |