Boundary Value Problems for Elliptic Systems

Contents......Page 8
Preface......Page 12
Index of Notation......Page 14
PART I: A Spectral Theory of Matrix Polynomials ......Page 16
1 Matrix polynomials ......Page 18
1.1 Smith canonical form ......Page 19
1.2 Eigenvectors and Jordan chains ......Page 23
1.3 Partial spectral pairs ......Page 27
1.4 Canonical set of Jordan chains ......Page 31
2 Spectral triples for matrix polynomials ......Page 38
2.1 Spectral triples ......Page 39
2.2 Properties of spectral triples and the Calderon projector ......Page 43
2.3 Left Jordan chains and a representation for the resolvent L-'(2) ......Page 50
2.4 Transformations of matrix polynomials ......Page 54
3.1 Standard pairs and triples ......Page 58
3.2 Linearization ......Page 62
3.3 Representation of a monic matrix polynomial in terms of a standard pair ......Page 66
3.4 Euclidean algorithm in terms of a standard pair ......Page 67
3.5 Monic divisors ......Page 72
3.6 Monic spectral divisors ......Page 73
3.7 Second degree matrix polynomials and examples ......Page 76
3.8 Changing from complex to real matrix coefficients ......Page 79
4 Further results ......Page 87
4.1 The inhomogeneous equation L(d/dt)u = f ......Page 88
4.2 Infinite spectral triples ......Page 90
4.3 More on restrictions of spectral pairs ......Page 98
4.4 Spectral triples of products ......Page 99
4.5 Transformations of products ......Page 106
PART II: Manifolds and Vector Bundles ......Page 114
5.1 Background and notation ......Page 116
5.2 Manifolds ......Page 128
5.3 The tangent bundle ......Page 134
5.4 Submanifolds ......Page 139
5.5 Vector fields ......Page 145
5.6 Partitions of unity ......Page 153
5.7 Vector bundles ......Page 157
5.8 Operations on vector bundles ......Page 167
5.9 Homotopy property for vector bundles ......Page 176
5.10 Riemannian and Hermitian metrics ......Page 179
5.11 Manifolds with boundary ......Page 183
5.12 Tubular neighbourhoods and collars ......Page 187
6.1 Differential forms ......Page 195
6.2 The exterior derivative d ......Page 197
6.3 The Poincar6 lemma ......Page 204
6.4 Orientation of a vector bundle ......Page 207
6.5 Orientation of a manifold ......Page 212
6.6 Integration on manifolds ......Page 216
6.7 Stokes' theorem ......Page 221
6.8 Differential operators in vector bundles ......Page 224
6.9 The Hodge star operator and the Laplace-deRham operator ......Page 232
PART III: Pseudo-Differential Operators and Elliptic Boundary Value Problems ......Page 246
7.1 Some remarks about generalizing integrals ......Page 248
7.2 The classes S' ......Page 251
7.3 Pseudo-differential operator algebra and asymptotics ......Page 258
7.4 Transformations of p.d.o.'s under a diffeomorphism ......Page 267
7.5 Classical symbols ......Page 271
7.6 Continuity in Sobolev spaces ......Page 275
7.7 Elliptic operators on R" ......Page 283
7.8 Girding's inequality and some results on the relation between the operator norm of a p.d.o. and the norm of its symbol ......Page 286
Appendix: summary of definitions and theorems for Sobolev spaces ......Page 293
8 Pseudo-differential operators on a compact manifold ......Page 302
8.1 Background and notation ......Page 303
8.2 Pseudo-differential operators on M ......Page 308
8.3 Main symbols and p.d.o. algebra ......Page 313
8.4 Classic operators on M ......Page 322
8.5 Definitions for operators in vector bundles ......Page 324
8.6 Pseudo-differential operators in vector bundles ......Page 328
8.7 Elliptic operators ......Page 339
8.8 An illustration: the Hodge decomposition theorem ......Page 350
8.9 Limits of pseudo-differential operators ......Page 353
8.10 The index of elliptic symbols ......Page 359
9.1 Fredholm operators and some functional analysis ......Page 380
9.2 Elliptic systems of Douglis-Nirenberg type ......Page 391
9.3 Boundary operators and the L-condition ......Page 402
9.4 The main theorem for elliptic boundary value problems ......Page 408
PART IV: Reduction of a Boundary Value Problem to an Elliptic System on the Boundary ......Page 428
10 Understanding the L-condition ......Page 430
10.1 Alternative versions of the L-condition ......Page 431
10.2 The Dirichlet problem ......Page 438
10.3 Matrix polynomials depending on parameters ......Page 442
10.4 Homogeneity properties of spectral triples ......Page 446
10.5 The classes Ell'-` and BE','-': two theorems of Agranovic and Dynin ......Page 454
10.6 Homotopies of elliptic boundary problems ......Page 458
10.7 The classes ((' and !ftl,' ......Page 462
10.8 Comparing the index of two problems having the same boundary operator ......Page 468
10.9 Composition of boundary problems ......Page 472
11 Applications to the index ......Page 480
11.1 First-order elliptic systems ......Page 481
11.2 Higher-order elliptic systems ......Page 490
12 BVP's for ordinary differential operators and the connection with spectral triples ......Page 502
12.1 Extension of C? functions defined on a half-line ......Page 503
12.2 Ordinary differential operators on a half-line ......Page 506
13.1 C' functions defined on a half-space ......Page 517
13.2 The transmission property ......Page 525
13.3 Boundary values of a single-layer potential ......Page 532
14 The main theorem revisited ......Page 540
14.1 Some spaces of distributions on R",. ......Page 545
14.2 The spaces H''' ......Page 554
14.3 The Calderon operator for an elliptic operator ......Page 559
14.4 Parametrix for an elliptic boundary problem ......Page 564
14.5 An application to the index ......Page 572
14.6 The main theorem for operators in '.8{'E`,', and the classes sm.m' ......Page 577
PART V: An Index Formula for Elliptic Boundary Problems in the Plane ......Page 592
15.1 Some preliminaries ......Page 594
15.2 The degree or winding number on the unit circle ......Page 597
15.3 The topological index ......Page 600
15.4 Changing from complex to real matrix coefficients ......Page 606
16 The index in the plane ......Page 610
16.1 A simple form for first-order elliptic systems with real coefficients ......Page 611
16.2 The index formula for first-order elliptic systems with real coefficients ......Page 615
16.3 A fundamental solution for first-order elliptic systems with constant real coefficients ......Page 617
16.4 Index formulas for higher-order systems with real coefficients ......Page 620
16.5 The index formula for elliptic systems with complex coefficients and when the boundary operator is pseudodifferential ......Page 630
17.1 Homotopy classification ......Page 638
17.2 An example: the Neumann BVP for second-order elliptic operators ......Page 642
17.3 An example: the elliptic system for plane elastic deformations ......Page 645
References ......Page 650
Index ......Page 654