Elliptic mixed, transmission and singular crack problems:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Zürich
European Mathematical Society Publishing House
2008
|
| Ausgabe: | 1. Aufl. |
| Schriftenreihe: | EMS tracts in mathematics
4 |
| Schlagworte: | |
| Online-Zugang: | Inhaltstext Inhaltstext Inhaltsverzeichnis |
| Beschreibung: | XII, 765 S. |
| ISBN: | 9783037190401 303719040X |
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| 100 | 1 | |a Harutyunyan, Gohar |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Elliptic mixed, transmission and singular crack problems |c Gohar Harutyunyan ; B. Wolfgang Schulze |
| 250 | |a 1. Aufl. | ||
| 264 | 1 | |a Zürich |b European Mathematical Society Publishing House |c 2008 | |
| 300 | |a XII, 765 S. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 1 | |a EMS tracts in mathematics |v 4 | |
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Datensatz im Suchindex
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Titel: Elliptic mixed, transmission and singular crack problems
Autor: Harutyunyan, Gohar
Jahr: 2008
Contents
Preface v
Introduction 1
1 Boundary value problems with mixed and interface data 13
1.1 Elliptic boundary value problems 13
1.1.1 Differential operators with classical boundary conditions . 13
1.1.2 The Poisson kernels in the half-space 18
1.1.3 Reduction to the boundary 21
1.1.4 Examples 26
1.2 Mixed and transmission problems 30
1.2.1 Mixed problems in weighted edge spaces 30
1.2.2 Additional conditions at the interface 34
1.2.3 Examples 42
1.2.4 Reduction of orders and reduction to the boundary 45
1.2.5 Mixed problems in standard Sobolev spaces 49
1.3 Problems with several types of interfaces 52
1.3.1 Transmission problems with smooth interfaces 52
1.3.2 Transmission problems with singular interfaces 54
2 Symbolic structures and associated operators 57
2.1 Scalar pseudo-differential calculus 57
2.1.1 Spaces of symbols and basic operations 57
2.1.2 Pseudo-differential operators and Sobolev spaces 61
2.1.3 Operators on manifolds 65
2.2 Calculus with operator-valued symbols 72
2.2.1 Symbols and operators with twisted homogeneity 73
2.2.2 Abstract edge spaces 79
2.2.3 Elements of the calculus 87
2.3 Operators on manifolds with conical exit to infinity 102
2.3.1 Symbols with exit behaviour 102
2.3.2 Operators globally in the Euclidean space 106
2.3.3 Operators on manifolds 107
2.3.4 Ellipticity in the scalar case Ill
2.3.5 Calculus with operator-valued symbols 112
2.4 Mellin operators 120
2.4.1 The Mellin transform 120
2.4.2 Weighted Sobolev spaces 122
2.4.3 Degenerate differential operators 132
2.4.4 Mellin operators and quantisation 137
viii Contents
2.4.5 A connection between edge-degenerate operators
and exit calculus 146
2.4.6 Green operators for conical singularities 155
3 Boundary value problems with the transmission property 167
3.1 Interior and boundary symbols 167
3.1.1 Symbols with the transmission property 167
3.1.2 Operators with the transmission property 173
3.1.3 Green operators on the half-axis 176
3.1.4 Boundary value problems on the half-axis 180
3.1.5 Operators on an interval 185
3.2 The algebra of boundary value problems 187
3.2.1 Global smoothing operators 188
3.2.2 Green operators 189
3.2.3 Boundary value problems 192
3.3 Ellipticity and parametrices 200
3.3.1 Elliptic boundary value problems 200
3.3.2 Parametrices and inverses 202
3.3.3 Parameter-dependent ellipticity 205
3.3.4 Fredholm families and block matrix isomorphisms 207
3.4 The calculus on manifolds with conical exit to infinity 214
3.4.1 Motivation in terms of principal edge symbols 214
3.4.2 Global operators in the half-space 217
3.4.3 Operators on a manifold with conical exit 221
3.4.4 A relation between edge-degenerate families and exit calculus . 224
4 Mixed problems in standard Sobolev spaces 225
4.1 Reductions of orders on a manifold with boundary 225
4.1.1 Order reducing symbols in the half-space 225
4.1.2 Actions in Sobolev spaces 228
4.1.3 A relationship with classical Volterra symbols 230
4.1.4 Global reduction of orders 233
4.1.5 General operators with plus/minus-symbols 235
4.2 Mixed elliptic problems 240
4.2.1 Mixed problems for differential operators 241
4.2.2 Ellipticity with additional conditions at the interface 257
4.2.3 The Zaremba problem 263
4.2.4 Jumping oblique derivatives and other examples 266
5 Mixed problems in weighted edge spaces 271
5.1 Mixed problems in edge spaces 271
5.1.1 Basic observations 271
5.1.2 Green symbols 274
5.1.3 The Zaremba problem as an edge problem 275
Contents ix
5.2 Relations between edge and standard Sobolev spaces 279
5.2.1 Spaces on the boundary 279
5.2.2 Edge spaces in the stretched domain 281
5.2.3 A reformulation of mixed problems from standard
Sobolev spaces 283
5.3 Elliptic interface conditions 287
5.3.1 Mixed problems in spaces of arbitrary weights 287
5.3.2 Construction of elliptic interface conditions 289
5.3.3 Parametrices and regularity of solutions for the Zaremba
problem 291
5.3.4 Jumping oblique derivatives and other examples 292
5.4 Edge calculus, specified to mixed problems 298
5.4.1 Edge amplitude functions 298
5.4.2 Edge-boundary value problems 301
5.4.3 Ellipticity and parametrices 305
5.4.4 Asymptotics of solutions 306
6 Operators on manifolds with conical singularities and boundary 309
6.1 Fuchs type operators and Mellin quantisation 309
6.1.1 Mellin quantisation 309
6.1.2 Kernel cut-off 323
6.1.3 Meromorphic Fredholm families and ellipticity of conormal
symbols 332
6.1.4 Green operators 338
6.1.5 Mellin operators with smoothing symbols 344
6.1.6 Operators with holomorphic Mellin symbols 346
6.2 The cone algebra 355
6.2.1 Operators on a compact manifold with conical singularities
and boundary 356
6.2.2 Operators on an infinite cone with boundary 363
6.3 Boundary value problems in plane domains 369
6.3.1 The Dirichlet problem in a strip 369
6.3.2 The Neumann and the Zaremba problem in a strip 373
6.3.3 The Dirichlet problem in an angle 376
6.3.4 The Neumann and the Zaremba problem in an angle 378
6.4 Special operators of the cone calculus 380
6.4.1 Reduction of orders 380
6.4.2 Operators on a cone with arbitrary weights at infinity 385
6.4.3 Cone operators with parameters 389
6.4.4 Elliptic regularity for some Schrodinger equation 391
x Contents
7 Operators on manifolds with edges and boundary 394
7.1 Differential operators on manifolds with edges 394
7.1.1 Edge-degenerate differential operators 394
7.1.2 Weighted edge spaces 397
7.1.3 Edge-boundary value problems as operators in weighted spaces 404
7.1.4 Operators in alternative weighted edge spaces 406
7.2 The edge algebra 408
7.2.1 Edge-degenerate symbols and operator conventions 409
7.2.2 Global smoothing operators 413
7.2.3 Green and smoothing Mellin symbols 416
7.2.4 Edge amplitude functions 420
7.2.5 The edge algebra 428
7.2.6 Ellipticity and reductions of orders 434
7.3 Mellin-edge representations of elliptic operators 442
7.3.1 Decomposition of classical Sobolev spaces 442
7.3.2 Edge decompositions of differential operators 447
7.3.3 Global constructions 451
7.3.4 Edge representation of boundary value problems 457
7.3.5 Relative index results 460
7.3.6 Interface conditions for small weights 463
7.4 The Laplacian in a wedge, and other elliptic operators of the edge
calculus 464
7.4.1 The Dirichlet problem in a wedge 464
7.4.2 The Neumann and the Zaremba problem in a wedge 467
7.4.3 Other examples of elliptic edge operators 468
8 Corner operators and problems with singular interfaces 475
8.1 Singular mixed problems and corner manifolds 475
8.1.1 The singular Zaremba problem 475
8.1.2 Operators in edge representation 477
8.1.3 Principal symbols and edge conditions 479
8.1.4 Corner manifolds 481
8.2 Corner operators in spaces with double weights 484
8.2.1 Transformation to a corner boundary value problem 484
8.2.2 Corner spaces with double weights 486
8.2.3 Continuity in corner spaces 491
8.2.4 Holomorphic corner symbols 495
8.2.5 Corner boundary value problems 497
8.2.6 Ellipticity near the corner 501
8.3 Corner edge operators 503
8.3.1 Global corner boundary value problems 503
8.3.2 Ellipticity and parametrices 508
8.3.3 The singular Zaremba problem 509
8.3.4 Remarks 512
Contents xi
8.4 Cracks with singularities at the boundary 514
8.4.1 Crack problems as edge-comer boundary value problems . 514
8.4.2 Operators near the smooth part of the crack boundary 519
8.4.3 Parameter-dependent crack operators on a sphere 522
8.4.4 The local corner-crack calculus 526
8.4.5 Singular crack problems 530
8.4.6 Examples 532
8.4.7 Further comments 537
8.5 Mixed problems with singular interfaces 538
8.5.1 Mixed problems in an infinite cylinder 538
8.5.2 Reduction to the boundary 541
8.5.3 Ellipticity with interface conditions 545
9 Operators in infinite cylinders and the relative index 552
9.1 Calculus with operator-valued meromorphic families 552
9.1.1 Characteristic values and factorisation of meromorphic families 552
9.1.2 The inhomogeneous equation 559
9.1.3 An index formula 566
9.2 Boundary value problems in infinite cylinders 567
9.2.1 Operators in cylindrical Sobolev spaces 567
9.2.2 Characteristic values and factorisation of meromorphic families 571
9.2.3 The relative index 573
9.3 The relative index for corner singularities 576
9.3.1 Parameter-dependent cone calculus 576
9.3.2 Meromorphic families 581
9.3.3 Examples 587
9.3.4 Characteristic values and factorisation 590
9.3.5 Operators on the infinite cylinder 592
9.3.6 The relative index 597
9.4 Cutting and pasting of elliptic operators 597
9.4.1 The locality of the index in the smooth case 598
9.4.2 Operators in bottleneck spaces 599
9.4.3 A general locality principle for the index 602
10 Intuitive ideas of the calculus on singular manifolds 604
10.1 Simple questions, unexpected answers 604
10.1.1 What is ellipticity? 604
10.1.2 Meromorphic symbolic structures 615
10.1.3 Naive and edge definitions of Sobolev spaces 623
10.2 Are regular boundaries harmless? 630
10.2.1 What is a boundary value problem? 631
10.2.2 Quantisation 646
10.2.3 The conormal cage 656
10.3 How interesting are conical singularities? 661
xii Contents
10.3.1 The iterative construction of higher singularities 662
10.3.2 Operators with sleeping parameters 666
10.3.3 Smoothing operators may contribute to the index 669
10.3.4 Are cylinders the genuine corners? 672
10.4 Is 'degenerate'bad? 674
10.4.1 Operators on stretched spaces 675
10.4.2 What is 'smoothness' on a singular manifold? 678
10.4.3 Schwartz kernels and Green operators 680
10.4.4 Pseudo-differential aspects, solvability of equations 682
10.4.5 Discrete, branching, and continuous asymptotics 686
10.5 Higher generations of calculi 694
10.5.1 Higher generations of weighted corner spaces 695
10.5.2 Additional edge conditions in higher corner algebras 699
10.5.3 A hierarchy of topological obstructions 701
10.5.4 The building of singular algebras 703
10.6 Historical background and future program 707
10.6.1 Achievements of the past development 707
10.6.2 Conification and edgification 710
10.6.3 Similarities and differences between ellipticity and parabolicity 714
10.6.4 Open problems and new challenges 719
10.6.5 Concluding remarks 723
Bibliography 729
List of Symbols 743
Index 751 |
| any_adam_object | 1 |
| author | Harutyunyan, Gohar Schulze, B. Wolfgang |
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| physical | XII, 765 S. |
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| series2 | EMS tracts in mathematics |
| spelling | Harutyunyan, Gohar Verfasser aut Elliptic mixed, transmission and singular crack problems Gohar Harutyunyan ; B. Wolfgang Schulze 1. Aufl. Zürich European Mathematical Society Publishing House 2008 XII, 765 S. txt rdacontent n rdamedia nc rdacarrier EMS tracts in mathematics 4 Randwertproblem (DE-588)4048395-2 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Elliptischer Pseudodifferentialoperator (DE-588)4309446-6 gnd rswk-swf Randwertproblem (DE-588)4048395-2 s Elliptische Differentialgleichung (DE-588)4014485-9 s Elliptischer Pseudodifferentialoperator (DE-588)4309446-6 s DE-604 Schulze, B. Wolfgang Verfasser aut EMS tracts in mathematics 4 (DE-604)BV022480257 4 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3029402&prov=M&dok_var=1&dok_ext=htm Inhaltstext text/html http://www.ems-ph.org/book.php?proj_nr=64&searchterm= Inhaltstext HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020132119&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Harutyunyan, Gohar Schulze, B. Wolfgang Elliptic mixed, transmission and singular crack problems EMS tracts in mathematics Randwertproblem (DE-588)4048395-2 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Elliptischer Pseudodifferentialoperator (DE-588)4309446-6 gnd |
| subject_GND | (DE-588)4048395-2 (DE-588)4014485-9 (DE-588)4309446-6 |
| title | Elliptic mixed, transmission and singular crack problems |
| title_auth | Elliptic mixed, transmission and singular crack problems |
| title_exact_search | Elliptic mixed, transmission and singular crack problems |
| title_full | Elliptic mixed, transmission and singular crack problems Gohar Harutyunyan ; B. Wolfgang Schulze |
| title_fullStr | Elliptic mixed, transmission and singular crack problems Gohar Harutyunyan ; B. Wolfgang Schulze |
| title_full_unstemmed | Elliptic mixed, transmission and singular crack problems Gohar Harutyunyan ; B. Wolfgang Schulze |
| title_short | Elliptic mixed, transmission and singular crack problems |
| title_sort | elliptic mixed transmission and singular crack problems |
| topic | Randwertproblem (DE-588)4048395-2 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Elliptischer Pseudodifferentialoperator (DE-588)4309446-6 gnd |
| topic_facet | Randwertproblem Elliptische Differentialgleichung Elliptischer Pseudodifferentialoperator |
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| volume_link | (DE-604)BV022480257 |
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