Elliptic Mixed, Transmission and Singular Crack Problems (EMS Tracts in Mathematics)

Preface......Page 5
Contents......Page 7
Introduction......Page 13
Differential operators with classical boundary conditions......Page 25
The Poisson kernels in the half-space......Page 30
Reduction to the boundary......Page 33
Examples......Page 38
Mixed problems in weighted edge spaces......Page 42
Additional conditions at the interface......Page 46
Examples......Page 54
Reduction of orders and reduction to the boundary......Page 57
Mixed problems in standard Sobolev spaces......Page 61
Transmission problems with smooth interfaces......Page 64
Transmission problems with singular interfaces......Page 66
Spaces of symbols and basic operations......Page 69
Pseudo-differential operators and Sobolev spaces......Page 73
Operators on manifolds......Page 77
Calculus with operator-valued symbols......Page 84
Symbols and operators with twisted homogeneity......Page 85
Abstract edge spaces......Page 91
Elements of the calculus......Page 99
Symbols with exit behaviour......Page 114
Operators globally in the Euclidean space......Page 118
Operators on manifolds......Page 119
Ellipticity in the scalar case......Page 123
Calculus with operator-valued symbols......Page 124
The Mellin transform......Page 132
Weighted Sobolev spaces......Page 134
Degenerate differential operators......Page 144
Mellin operators and quantisation......Page 149
A connection between edge-degenerate operators and exit calculus......Page 158
Green operators for conical singularities......Page 167
Symbols with the transmission property......Page 179
Operators with the transmission property......Page 185
Green operators on the half-axis......Page 188
Boundary value problems on the half-axis......Page 192
Operators on an interval......Page 197
The algebra of boundary value problems......Page 199
Global smoothing operators......Page 200
Green operators......Page 201
Boundary value problems......Page 204
Elliptic boundary value problems......Page 212
Parametrices and inverses......Page 214
Parameter-dependent ellipticity......Page 217
Fredholm families and block matrix isomorphisms......Page 219
Motivation in terms of principal edge symbols......Page 226
Global operators in the half-space......Page 229
Operators on a manifold with conical exit......Page 233
A relation between edge-degenerate families and exit calculus......Page 236
Order reducing symbols in the half-space......Page 237
Actions in Sobolev spaces......Page 240
A relationship with classical Volterra symbols......Page 242
Global reduction of orders......Page 245
General operators with plus/minus-symbols......Page 247
Mixed elliptic problems......Page 252
Mixed problems for differential operators......Page 253
Ellipticity with additional conditions at the interface......Page 269
The Zaremba problem......Page 275
Jumping oblique derivatives and other examples......Page 278
Basic observations......Page 283
Green symbols......Page 286
The Zaremba problem as an edge problem......Page 287
Spaces on the boundary......Page 291
Edge spaces in the stretched domain......Page 293
A reformulation of mixed problems from standard Sobolev spaces......Page 295
Mixed problems in spaces of arbitrary weights......Page 299
Construction of elliptic interface conditions......Page 301
Parametrices and regularity of solutions for the Zaremba problem......Page 303
Jumping oblique derivatives and other examples......Page 304
Edge amplitude functions......Page 310
Edge-boundary value problems......Page 313
Ellipticity and parametrices......Page 317
Asymptotics of solutions......Page 318
Mellin quantisation......Page 321
Kernel cut-off......Page 335
Meromorphic Fredholm families and ellipticity of conormal symbols......Page 344
Green operators......Page 350
Mellin operators with smoothing symbols......Page 356
Operators with holomorphic Mellin symbols......Page 358
The cone algebra......Page 367
Operators on a compact manifold with conical singularities and boundary......Page 368
Operators on an infinite cone with boundary......Page 375
The Dirichlet problem in a strip......Page 381
The Neumann and the Zaremba problem in a strip......Page 385
The Dirichlet problem in an angle......Page 388
The Neumann and the Zaremba problem in an angle......Page 390
Reduction of orders......Page 392
Operators on a cone with arbitrary weights at infinity......Page 397
Cone operators with parameters......Page 401
Elliptic regularity for some Schrödinger equation......Page 403
Edge-degenerate differential operators......Page 406
Weighted edge spaces......Page 409
Edge-boundary value problems as operators in weighted spaces......Page 416
Operators in alternative weighted edge spaces......Page 418
The edge algebra......Page 420
Edge-degenerate symbols and operator conventions......Page 421
Global smoothing operators......Page 425
Green and smoothing Mellin symbols......Page 428
Edge amplitude functions......Page 432
The edge algebra......Page 440
Ellipticity and reductions of orders......Page 446
Decomposition of classical Sobolev spaces......Page 454
Edge decompositions of differential operators......Page 459
Global constructions......Page 463
Edge representation of boundary value problems......Page 469
Relative index results......Page 472
Interface conditions for small weights......Page 475
The Dirichlet problem in a wedge......Page 476
The Neumann and the Zaremba problem in a wedge......Page 479
Other examples of elliptic edge operators......Page 480
The singular Zaremba problem......Page 487
Operators in edge representation......Page 489
Principal symbols and edge conditions......Page 491
Corner manifolds......Page 493
Transformation to a corner boundary value problem......Page 496
Corner spaces with double weights......Page 498
Continuity in corner spaces......Page 503
Holomorphic corner symbols......Page 507
Corner boundary value problems......Page 509
Ellipticity near the corner......Page 513
Global corner boundary value problems......Page 515
Ellipticity and parametrices......Page 520
The singular Zaremba problem......Page 521
Remarks......Page 524
Crack problems as edge-corner boundary value problems......Page 526
Operators near the smooth part of the crack boundary......Page 531
Parameter-dependent crack operators on a sphere......Page 534
The local corner-crack calculus......Page 538
Singular crack problems......Page 542
Examples......Page 544
Further comments......Page 549
Mixed problems in an infinite cylinder......Page 550
Reduction to the boundary......Page 552
Ellipticity with interface conditions......Page 557
Characteristic values and factorisation of meromorphic families......Page 564
The inhomogeneous equation......Page 571
An index formula......Page 578
Operators in cylindrical Sobolev spaces......Page 579
Characteristic values and factorisation of meromorphic families......Page 583
The relative index......Page 585
Parameter-dependent cone calculus......Page 588
Meromorphic families......Page 593
Examples......Page 599
Characteristic values and factorisation......Page 602
Operators on the infinite cylinder......Page 604
Cutting and pasting of elliptic operators......Page 609
The locality of the index in the smooth case......Page 610
Operators in bottleneck spaces......Page 611
A general locality principle for the index......Page 614
What is ellipticity?......Page 616
Meromorphic symbolic structures......Page 627
Naive and edge definitions of Sobolev spaces......Page 635
Are regular boundaries harmless?......Page 642
What is a boundary value problem?......Page 643
Quantisation......Page 658
The conormal cage......Page 667
How interesting are conical singularities?......Page 673
The iterative construction of higher singularities......Page 674
Operators with sleeping parameters......Page 678
Smoothing operators may contribute to the index......Page 681
Are cylinders the genuine corners?......Page 684
Is `degenerate\' bad?......Page 686
Operators on stretched spaces......Page 687
What is `smoothness\' on a singular manifold?......Page 690
Schwartz kernels and Green operators......Page 692
Pseudo-differential aspects, solvability of equations......Page 694
Discrete, branching, and continuous asymptotics......Page 698
Higher generations of calculi......Page 706
Higher generations of weighted corner spaces......Page 707
Additional edge conditions in higher corner algebras......Page 711
A hierarchy of topological obstructions......Page 713
The building of singular algebras......Page 715
Achievements of the past development......Page 719
Conification and edgification......Page 722
Similarities and differences between ellipticity and parabolicity......Page 726
Open problems and new challenges......Page 731
Concluding remarks......Page 735
Bibliography......Page 741
List of Symbols......Page 755
Index......Page 763