Aspects Of Boundary Problems In Analysis And Geometry

by Gil, Juan

• ISBN13: 9783764370695
• ISBN10: 3764370696
• Format: Hardcover
• Copyright: 2004-05-14
• Publisher: Birkhauser
• More Book Details

Summary

Boundary problems constitute an essential field of common mathematical interest. The intention of this volume is to highlight several analytic and geometric aspects of boundary problems with special emphasis on their interplay. It includes surveys on classical topics presented from a modern perspective as well as reports on current research. The collection splits into two related groups: - analysis and geometry of geometric operators and their index theory - elliptic theory of boundary value problems and the Shapiro-Lopatinsky condition

Table of Contents

Preface	xi
Part I. Geometric Operators and the Index
Spectral invariants of operators of Dirac type on partitioned manifolds
by DAVID BLEECKER AND BERNHELM BOOSS-BAVNBEK
Introduction	1	(130)
1. Basic notations and results	3	(25)
1.1. Index of Fredholm operators and spectral flow of curves of self-adjoint Fredholm operators	3	(9)
1.2. Symmetric operators and symplectic analysis	12	(7)
1.3. Operators of Dirac type and their ellipticity	19	(1)
1.4. Weak unique continuation property	20	(8)
2. The index of elliptic operators on partitioned manifolds	28	(51)
2.1. Examples and the Hellwig-Vekua index theorem	28	(10)
2.2. The index of twisted Dirac operators on closed manifolds	38	(14)
2.3. Dirac type operators on manifolds with boundary	52	(10)
2.4. The Atiyah-Patodi-Singer index theorem	62	(11)
2.5. Symplectic geometry of Cauchy data spaces	73	(3)
2.6. Non-additivity of the index	76	(2)
2.7. Pasting of spectral flow	78	(1)
3. The eta invariant	79	(47)
3.1. Functional integrals and spectral asymmetry	83	(3)
3.2. The ζ-determinant for operators of infinite rank	86	(2)
3.3. Spectral invariants of different 'sensitivity'	88	(2)
3.4. Pasting formulas for the eta invariant - outlines	90	(6)
3.5. The adiabatic additivity of the small-t chopped η-invariant	96	(7)
3.6. Asymptotic vanishing of the large-t chopped η-invariant on stretched part manifold	103	(6)
3.7. The estimate of the lowest nontrivial eigenvalue	109	(6)
3.8. The spectrum on the closed stretched manifold	115	(7)
3.9. The additivity for spectral boundary conditions	122	(4)
References	126	(5)
Index theory of Dirac operators on manifolds with corners up to codimension two
by PAUL LOYA
1. Introduction: The Gauss-Bonnet formula and index theory	131	(4)
1.1. The classical Gauss-Bonnet formula	132	(1)
1.2. The Gauss-Bonnet formula as an index formula	133	(2)
2. The Atiyah-Singer index formula	135	(5)
2.1. Statement of the Atiyah-Singer index theorem	135	(3)
2.2. Outline of the proof of the Atiyah-Singer formula	138	(2)
2.3. Some remarks on the Atiyah-Singer index theorem	140	(1)
3. The Atiyah-Patodi-Singer index formula	140	(8)
3.1. Attaching a cylindrical end	141	(3)
3.2. Statement of the Atiyah-Patodi-Singer index theorem	144	(2)
3.3. Interpretation as b-objects	146	(1)
3.4. Some remarks on the Atiyah-Patodi-Singer index theorem	147	(1)
4. Melrose's b-geometry proof of the Atiyah-Patodi-Singer theorem	148	(5)
4.1. The proof of APS with details left out	148	(2)
4.2. Some facts about the heat kernels	150	(1)
4.3. Filling in the details for the b-trace	150	(2)
4.4. Filling in the details for the eta invariant	152	(1)
5. Index theory on manifolds with corners of codimension two	153	(8)
5.1. Dirac operators on manifolds with corners	154	(2)
5.2. Attaching multi-cylindrical ends	156	(2)
5.3. Muller's generalization of the APS index formula	158	(1)
5.4. b-version of Muller's theorem	159	(1)
5.5. Some remarks on index theory on manifolds with corners	160	(1)
6. Perturbations of Dirac operators on manifolds with corners	161	(5)
6.1. Fredholm perturbation of Dirac operators	161	(2)
6.2. An index formula for perturbed Dirac operators	163	(2)
6.3. Some concluding remarks	165	(1)
References	166	(4)
Index defects in the theory of spectral boundary value problems
by ANTON SAVIN AND BORIS STERNIN
Introduction	170	(12)
1. Spectral boundary value problems and their index	182	(8)
1.1. Atiyah-Patodi-Singer spectral boundary value problems	182	(1)
1.2. The spectral flow	183	(3)
1.3. A theorem on index decompositions	186	(3)
1.4. Examples	189	(1)
2. Index defects for problems with parity conditions	190	(8)
2.1. The dimension functional for even subspaces	190	(3)
2.2. The index defect formula	193	(3)
2.3. The dimension functional and the η-invariant	196	(2)
3. Index defects on twisted Zn-manifolds	198	(14)
3.1. Twisted Zn-manifolds and elliptic operators	198	(3)
3.2. The Freed-Melrose index theorem modulo n	201	(2)
3.3. The index defect problem on twisted Zn-manifolds	203	(2)
3.4. The element of K-theory defined by a manifold whose boundary is a covering	205	(2)
3.5. The index defect formula	207	(4)
3.6. An application to η-invariants	211	(1)
4. Appendix A. The Atiyah-Patodi-Singer η-invariant	212	(5)
4.1. The geometric index formula and the η-invariant	212	(2)
4.2. The derivative of the η-invariant	214	(1)
4.3. The homotopy invariance of the η-invariant	214	(3)
5. Appendix B. Elliptic operators and Poincaré duality. Smooth theory	217	(8)
5.1. The Poincaré isomorphism on a closed manifold	218	(4)
5.2. Duality and the topological index	222	(1)
5.3. Poincaré duality on manifolds with boundary. Absolute and relative cycles	223	(2)
6. Appendix C. Poincaré duality on Zn-manifolds	225	(10)
6.1. Relative cycles. The C*-algebra of a Zn-manifold	226	(2)
6.2. Absolute cycles. Nonlocal operators	228	(3)
6.3. The Poincaré isomorphism	231	(1)
6.4. Poincaré duality	232	(2)
6.5. A topological index for Zn-manifolds	234	(1)
References	235	(4)
Cyclic homology and pseudodifferential operators, a survey
by MOULAY-TAHAR BENAMEUR, JACEK BRODZKI, AND VICTOR NISTOR
Introduction	239	(26)
1. Hochschild homology	240	(5)
2. Cyclic homology	245	(7)
3. Introducing topology	252	(2)
4. Algebras of pseudodifferential operators	254	(5)
5. Applications and examples	259	(1)
5.1. Manifolds with corners	259	(1)
5.2. Fibrations by manifolds with corners	260	(1)
5.3. Longitudinal symbols on foliations	260	(2)
References	262	(3)
Index and secondary index theory for flat bundles with duality
by ULRICH BUNKE AND XIAONAN MA
1. Introduction to the paper	265	(2)
2. The functor Lepsilon	267	(13)
2.1. Introduction and summary	267	(1)
2.2. Definition and first properties	268	(3)
2.3. Computation of Lepsilon(X)	271	(4)
2.4. The natural transformation to Kappa-theory	275	(1)
2.5. Push-forward for Lepsilon	276	(4)
3. The functor Lepsilon	280	(16)
3.1. Introduction and summary	280	(1)
3.2. Secondary Kappa-theory	281	(1)
3.3. The functor LR/Z	282	(1)
3.4. The functor L	283	(10)
3.5. Functorial properties	293	(3)
4. Eta homomorphisms	296	(9)
4.1. Introduction and summary	296	(1)
4.2. η-1R/Z:-1R/Z(M)-->R/Z	296	(1)
4.3. R-valued η-homomorphisms	297	(2)
4.4. Definition of the extended L-groups	299	(5)
4.5. Construction of η:Lepsilonex(M) -->R	304	(1)
5. The secondary index map	305	(17)
5.1. Introduction and summary	305	(1)
5.2. Adiabatic limits of eta invariants, the eta form	306	(8)
5.3. Definition of the secondary index maps	314	(2)
5.4. Functorial properties	316	(3)
5.5. The index map for L epsilon ex and L epsilon ex	319	(3)
6. Adiabatic limits - sketches of proofs	322	(18)
6.1. Remarks	322	(1)
6.2. The proof of Theorem 4.17	322	(8)
6.3. Proof of Theorem 5.9	330	(3)
6.4. Proof of Theorem 5.10	333	(2)
6.5. Proof of Theorem 5.11	335	(5)
References	340	(2)
Part II. Elliptic Boundary Value Problems
Toeplitz operators, and ellipticity of boundary value problems with global projection conditions
by B.-W. SCHULZE
Introduction	342	(6)
1. Elements of the classical calculus of boundary value problems	348	(37)
1.1. Pseudo-differential and Toeplitz operators on a closed manifold	348	(21)
1.2. Operators with the transmission property at the boundary	369	(14)
1.3. SL-elliptic boundary value problems	383	(2)
2. Ellipticity with global projection conditions	385	(16)
2.1. The index obstruction	385	(7)
2.2. A Toeplitz algebra of boundary value problems	392	(3)
2.3. Ellipticity, parametrices, and the Fredholm property	395	(5)
2.4. Reduction to the boundary	400	(1)
3. Transmission operators and Cauchy data spaces	401	(21)
3.1. Transmission operators	401	(7)
3.2. Examples	408	(2)
3.3. Spectral boundary value problems	410	(10)
3.4. Calderon-Seeley projections	420	(2)
4. Remarks on the edge calculus with global projection data	422	(4)
4.1. Boundary value problems without the transmission property	422	(1)
4.2. Edge problems	423	(3)
4.3. Analysis on manifolds with singularities	426	(1)
References	426	(4)
On the tangential oblique derivative problem - methods, results, open problems
by PETER POPIVANOV
Introduction	430	(2)
1. Subelliptic operators and their application to the tangential oblique derivative problem	432	(25)
1.1. Subelliptic estimates	432	(10)
1.2. Local solvability and microlocal analysis	442	(5)
1.3. Canonical transformations	447	(2)
1.4. The Calderon projector	449	(3)
1.5. Hölder regularity	452	(5)
2. Survey on the degenerate oblique derivative problem in Hölder, Sobolev, and Besov spaces	457	(12)
2.1. Tangential vector fields and solvability	458	(6)
2.2. Modified oblique derivative problem	464	(5)
3. Open problems	469	(1)
References	470	(2)
A note on boundary value problems on manifolds with cylindrical ends
by MARIUS MITREA AND VICTOR NISTOR
Introduction	472	(3)
1. Operators on manifolds with cylindrical ends	475	(3)
1.1. Manifolds with cylindrical ends and the Laplace operator	475	(1)
1.2. Operators that are translation invariant in a neighborhood of infinity	476	(2)
2. A spectrally invariant algebra	478	(4)
2.1. Operators that are almost translation invariant in a neighborhood of infinity	479	(2)
2.2. The "Δ + V trick"	481	(1)
3. Boundary layer potential integrals	482	(3)
3.1. Submanifolds with cylindrical ends	482	(1)
3.2. Boundary layer potential integrals	483	(2)
4. Layer potentials depending on a parameter	485	(4)
5. The Dirichlet problem	489	(3)
5.1. The Dirichlet-to-Neumann map	492	(1)
References	492	(3)
Relative elliptic theory
by VLADIMIR NAZAIKINSKII AND BORIS STERNIN
Introduction	495	(9)
Analytic Aspects of Relative Elliptic Theory	504	(1)
1. Classical objects. Lagrangian formalism	505	(14)
1.1. Lagrangian manifolds with measure	505	(4)
1.2. Composition and transposition formulas	509	(2)
1.3. Symbols	511	(5)
1.4. The product of symbols (twisted multiplication)	516	(3)
2. Quantization and the algebra of operator morphisms	519	(6)
2.1. The main operators	519	(3)
2.2. Composition formulas and adjoint operators	522	(1)
2.3. Algebras of operator morphisms	523	(2)
3. Elliptic morphisms and the index formula	525	(9)
3.1. The Green equation	526	(3)
3.2. Ellipticity and the Fredholm property	529	(4)
3.3. The index theorem for elliptic morphisms	533	(1)
4. The Sobolev problem	534	(5)
4.1. Sobolev problems and elliptic morphisms	534	(2)
4.2. The Fredholm property	536	(1)
4.3. The Sobolev problem and the adjoint problem	537	(1)
4.4. The index formula	538	(1)
Topological Aspects of Relative Elliptic Theory	539	(3)
5. Preliminaries	539	(1)
6. The metric trace of an elliptic operator	539	(3)
7. The pullback of an elliptic operator	542	(2)
7.1. The pullback of a pseudodifferential operator	542	(1)
7.2. The regularized pullback	543	(1)
7.3. The Riemann-Roch theorem for the regularized pullback	544	(1)
8. A finitely regularized pullback	544	(6)
8.1. A counterexample	544	(2)
8.2. The pullback for linear principal symbols	546	(2)
8.3. Pullbacks and exterior tensor products	548	(2)
9. The pullback of geometric operators	550	(5)
9.1. Group operators	550	(1)
9.2. The Euler and Hirzebruch operators	551	(3)
9.3. The Todd and Dirac operators	554	(1)
Appendix. Fourier Integral Operators	555	(4)
A.l. Homogeneous Lagrangian manifolds	555	(1)
A.2. Local description of homogeneous Lagrangian manifolds	556	(1)
A.3. Composition of homogeneous Lagrangian manifolds	556	(1)
A.4. Definition of Fourier integral operators	556	(2)
A.5. Pseudodifferential operators as Fourier integral operators	558	(1)
A.6. Boundedness theorems	558	(1)
A.7. Composition theorems	558	(1)
A.8. L2-adjoints of Fourier integral operators	559	(1)
A.9. Fourier integral operators on sections of vector bundles	559	(1)
References	559