Boundary value problems for elliptic systems:
This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index theory by means of pseudo-differential operators and new...
Gespeichert in:
| Hauptverfasser: | , , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Cambridge u.a.
Cambridge Univ. Press
1995
|
| Ausgabe: | 1. publ. |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Zusammenfassung: | This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the reader to work efficiently with the principal symbols of the elliptic and boundary operators. It also leads to important simplifications and unifications in the proofs of basic theorems such as the reformulation of the Lopatinskii condition in various equivalent forms, homotopy lifting theorems, the reduction of a system with boundary conditions to a system on the boundary, and the index formula for systems in the plane This book is suitable for use in graduate level courses on partial differential equations, elliptic systems, pseudo-differential operators, and matrix analysis. All the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises |
| Beschreibung: | XIV, 641 S. |
| ISBN: | 0521430119 |
Internformat
MARC
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| 245 | 1 | 0 | |a Boundary value problems for elliptic systems |c J. T. Wloka ; B. Rowley ; B. Lawruk |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge u.a. |b Cambridge Univ. Press |c 1995 | |
| 300 | |a XIV, 641 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 520 | 3 | |a This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the reader to work efficiently with the principal symbols of the elliptic and boundary operators. It also leads to important simplifications and unifications in the proofs of basic theorems such as the reformulation of the Lopatinskii condition in various equivalent forms, homotopy lifting theorems, the reduction of a system with boundary conditions to a system on the boundary, and the index formula for systems in the plane | |
| 520 | |a This book is suitable for use in graduate level courses on partial differential equations, elliptic systems, pseudo-differential operators, and matrix analysis. All the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises | ||
| 650 | 7 | |a Equations différentielles elliptiques |2 ram | |
| 650 | 7 | |a Problèmes aux limites |2 ram | |
| 650 | 4 | |a Boundary value problems | |
| 650 | 4 | |a Differential equations, Elliptic | |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptisches System |0 (DE-588)4121184-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptisches Randwertproblem |0 (DE-588)4193399-0 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Randwertproblem |0 (DE-588)4048395-2 |D s |
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| 689 | 1 | 0 | |a Elliptisches Randwertproblem |0 (DE-588)4193399-0 |D s |
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| 689 | 2 | 0 | |a Randwertproblem |0 (DE-588)4048395-2 |D s |
| 689 | 2 | 1 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |D s |
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Datensatz im Suchindex
| _version_ | 1804124905888808960 |
|---|---|
| adam_text | Contents
Preface page xi
Index of Notation xiii
PART I: A Spectra] Theory of Matrix Polynomials page 1
1 Matrix polynomials 3
1.1 Smith canonical form 4
1.2 Eigenvectors and Jordan chains 8
1.3 Partial spectral pairs 12
1.4 Canonical set of Jordan chains 16
2 Spectral triples for matrix polynomials 23
2.1 Spectral triples 24
2.2 Properties of spectral triples and the Calderon projector 28
2.3 Left Jordan chains and a representation for the resolvent
L~ X) 35
2.4 Transformations of matrix polynomials 39
3 Monic matrix polynomials 43
3.1 Standard pairs and triples 43
3.2 Linearization 47
3.3 Representation of a monic matrix polynomial in terms of
a standard pair 51
3.4 Euclidean algorithm in terms of a standard pair 52
3.5 Monic divisors 57
3.6 Monic spectral divisors 58
3.7 Second degree matrix polynomials and examples 61
3.8 Changing from complex to real matrix coefficients 64
4 Further results 72
4.1 The inhomogeneous equation L(d/dt)u = f 73
4.2 Infinite spectral triples 75
4.3 More on restrictions of spectral pairs 83
4.4 Spectral triples of products 84
4.5 Transformations of products 91
vii
viii Contents
PART II: Manifolds and Vector Bundles 99
5 Manifolds and vector bundles 101
5.1 Background and notation 101
5.2 Manifolds 113
5.3 The tangent bundle 119
5.4 Submanifolds 124
5.5 Vector fields 130
5.6 Partitions of unity 138
5.7 Vector bundles 142
5.8 Operations on vector bundles 152
5.9 Homotopy property for vector bundles 161
5.10 Riemannian and Hermitian metrics 164
5.11 Manifolds with boundary 168
5.12 Tubular neighbourhoods and collars 172
6 Differential forms 180
6.1 Differential forms 180
6.2 The exterior derivative d 182
6.3 The Poincare lemma 189
6.4 Orientation of a vector bundle 192
6.5 Orientation of a manifold 197
6.6 Integration on manifolds 201
6.7 Stokes theorem 206
6.8 Differential operators in vector bundles 209
6.9 The Hodge star operator and the Laplace de Rham
operator 217
PART III: Pseudo Differential Operators and Elliptic Boundary
Value Problems 231
7 Pseudo differential operators on W 233
7.1 Some remarks about generalizing integrals 233
7.2 The classes Sm 236
7.3 Pseudo differential operator algebra and asymptotics 243
7.4 Transformations of p.d.o. s under a diffeomorphism 252
7.5 Classical symbols 256
7.6 Continuity in Sobolev spaces 260
7.7 Elliptic operators on W 268
7.8 Garding s inequality and some results on the relation
between the operator norm of a p.d.o. and the norm of its
symbol 271
Appendix: summary of definitions and theorems for Sobolev
spaces 278
Contents ix
8 Pseudo differential operators on a compact manifold 287
8.1 Background and notation 288
8.2 Pseudo differential operators on M 293
8.3 Main symbols and p.d.o. algebra 298
8.4 Classic operators on M 307
8.5 Definitions for operators in vector bundles 309
8.6 Pseudo differential operators in vector bundles 313
8.7 Elliptic operators 324
8.8 An illustration: the Hodge decomposition theorem 335
8.9 Limits of pseudo differential operators 338
8.10 The index of elliptic symbols 344
9 Elliptic systems on bounded domains in W 365
9.1 Fredholm operators and some functional analysis 365
9.2 Elliptic systems of Douglis Nirenberg type 376
9.3 Boundary operators and the L condition 387
9.4 The main theorem for elliptic boundary value problems 393
PART IV: Reduction of a Boundary Value Problem to an Elliptic
System on the Boundary 413
10 Understanding the /. condition 415
10.1 Alternative versions of the L condition 416
10.2 The Dirichlet problem 423
10.3 Matrix polynomials depending on parameters 427
10.4 Homogeneity properties of spectral triples 431
10.5 The classes Ellst and BEs tm: two theorems of Agranovic
and Dynin 439
10.6 Homotopies of elliptic boundary problems 443
10.7 The classes and % m 447
10.8 Comparing the index of two problems having the same
boundary operator 453
10.9 Composition of boundary problems 457
11 Applications to the index 465
11.1 First order elliptic systems 466
11.2 Higher order elliptic systems 475
12 BVP s for ordinary differential operators and the connection
with spectral triples 487
12.1 Extension of C°° functions defined on a half line 488
12.2 Ordinary differential operators on a half line 491
x Contents
13 Behaviour of a pseudo differential operator near a boundary 502
13.1 Cx functions denned on a half space 502
13.2 The transmission property 510
13.3 Boundary values of a single layer potential 517
14 The main theorem revisited 525
14.1 Some spaces of distributions on W+ 530
14.2 The spaces Hst 539
14.3 The Calderon operator for an elliptic operator 544
14.4 Parametrix for an elliptic boundary problem 549
14.5 An application to the index 557
14.6 The main theorem for operators in ©Cm, and the ;
classes Sm m 562
PART V: An Index Formula for Elliptic Boundary Problems in
the Plane 577
15 Further results on the Lopatinskii condition 579
15.1 Some preliminaries 579
15.2 The degree or winding number on the unit circle 582
15.3 The topological index 585 ,
15.4 Changing from complex to real matrix coefficients 591
16 The index in the plane 595
16.1 A simple form for first order elliptic systems with real
coefficients 596
16.2 The index formula for first order elliptic systems with real
coefficients 600
16.3 A fundamental solution for first order elliptic systems
with constant real coefficients 602
16.4 Index formulas for higher order systems with real coeffi¬
cients 605
16.5 The index formula for elliptic systems with complex coef¬
ficients and when the boundary operator is pseudo
differential 615
17 Elliptic systems with 2x2 real coefficients 623
17.1 Homotopy classification 623
17.2 An example: the Neumann BVP for second order elliptic
operators 627
17.3 An example: the elliptic system for plane elastic defor¬
mations 630
References 635
Index 639
|
| any_adam_object | 1 |
| author | Wloka, Joseph 1929- Rowley, B. Lawruk, B. |
| author_GND | (DE-588)136730108 |
| author_facet | Wloka, Joseph 1929- Rowley, B. Lawruk, B. |
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| dewey-full | 515/.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| indexdate | 2024-07-09T17:53:06Z |
| institution | BVB |
| isbn | 0521430119 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-006978689 |
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| physical | XIV, 641 S. |
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| spelling | Wloka, Joseph 1929- Verfasser (DE-588)136730108 aut Boundary value problems for elliptic systems J. T. Wloka ; B. Rowley ; B. Lawruk 1. publ. Cambridge u.a. Cambridge Univ. Press 1995 XIV, 641 S. txt rdacontent n rdamedia nc rdacarrier This book examines the theory of boundary value problems for elliptic systems of partial differential equations, a theory which has many applications in mathematics and the physical sciences. The aim is to simplify and to algebraize the index theory by means of pseudo-differential operators and new methods in the spectral theory of matrix polynomials. This latter theory provides important tools that will enable the reader to work efficiently with the principal symbols of the elliptic and boundary operators. It also leads to important simplifications and unifications in the proofs of basic theorems such as the reformulation of the Lopatinskii condition in various equivalent forms, homotopy lifting theorems, the reduction of a system with boundary conditions to a system on the boundary, and the index formula for systems in the plane This book is suitable for use in graduate level courses on partial differential equations, elliptic systems, pseudo-differential operators, and matrix analysis. All the theorems are proved in detail, and the methods are well illustrated through numerous examples and exercises Equations différentielles elliptiques ram Problèmes aux limites ram Boundary value problems Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Elliptisches System (DE-588)4121184-4 gnd rswk-swf Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Elliptisches System (DE-588)4121184-4 s Randwertproblem (DE-588)4048395-2 s DE-604 Elliptisches Randwertproblem (DE-588)4193399-0 s Elliptische Differentialgleichung (DE-588)4014485-9 s 1\p DE-604 Rowley, B. Verfasser aut Lawruk, B. Verfasser aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006978689&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Wloka, Joseph 1929- Rowley, B. Lawruk, B. Boundary value problems for elliptic systems Equations différentielles elliptiques ram Problèmes aux limites ram Boundary value problems Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptisches System (DE-588)4121184-4 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4048395-2 (DE-588)4121184-4 (DE-588)4193399-0 |
| title | Boundary value problems for elliptic systems |
| title_auth | Boundary value problems for elliptic systems |
| title_exact_search | Boundary value problems for elliptic systems |
| title_full | Boundary value problems for elliptic systems J. T. Wloka ; B. Rowley ; B. Lawruk |
| title_fullStr | Boundary value problems for elliptic systems J. T. Wloka ; B. Rowley ; B. Lawruk |
| title_full_unstemmed | Boundary value problems for elliptic systems J. T. Wloka ; B. Rowley ; B. Lawruk |
| title_short | Boundary value problems for elliptic systems |
| title_sort | boundary value problems for elliptic systems |
| topic | Equations différentielles elliptiques ram Problèmes aux limites ram Boundary value problems Differential equations, Elliptic Elliptische Differentialgleichung (DE-588)4014485-9 gnd Randwertproblem (DE-588)4048395-2 gnd Elliptisches System (DE-588)4121184-4 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| topic_facet | Equations différentielles elliptiques Problèmes aux limites Boundary value problems Differential equations, Elliptic Elliptische Differentialgleichung Randwertproblem Elliptisches System Elliptisches Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=006978689&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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