Elliptic Boundary Problems For Dirac Operators (mathematics: Theory & Applications)

Elliptic boundary problems have enjoyed interest recently, espe­ cially among C\* -algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec­ ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical con­ texts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial dif­ ferential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more pieces than does the elliptic theory on closed manifolds. But that is not the only reason. Front Matter....Pages i-xviii Front Matter....Pages 1-1 Clifford Algebras and Clifford Modules....Pages 3-9 Clifford Bundles and Compatible Connections....Pages 10-18 Dirac Operators....Pages 19-25 Dirac Laplacian and Connection Laplacian....Pages 26-28 Euclidean Examples....Pages 29-35 The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds....Pages 36-39 Dirac Operators and Chirality....Pages 40-42 Unique Continuation Property for Dirac Operators....Pages 43-49 Invertible Doubles....Pages 50-58 Glueing Constructions. Relative Index Theorem....Pages 59-63 Front Matter....Pages 65-65 Sobolev Spaces on Manifolds with Boundary....Pages 67-74 Calderón Projector for Dirac Operators....Pages 75-94 Existence of Traces of Null Space Elements....Pages 95-104 Spectral Projections of Dirac Operators....Pages 105-110 Pseudo-Differential Grassmannians....Pages 111-126 The Homotopy Groups of the Space of Self-Adjoint Fredholm Operators....Pages 127-137 The Spectral Flow of Families of Self-Adjoint Operators....Pages 138-160 Front Matter....Pages 161-161 Elliptic Boundary Problems and Pseudo-Differential Projections....Pages 163-179 Regularity of Solutions of Elliptic Boundary Problems....Pages 180-187 Fredholm Property of the Operator A R ....Pages 188-204 Front Matter....Pages 161-161 Exchanges on the Boundary: Agranovič-Dynin Type Formulas and the Cobordism Theorem for Dirac Operators....Pages 205-210 The Index Theorem for Atiyah-Patodi-Singer Problems....Pages 211-252 Some Remarks on the Index of Generalized Atiyah-Patodi-Singer Problems....Pages 253-261 Bojarski’s Theorem. General Linear Conjugation Problems....Pages 262-275 Cutting and Pasting of Elliptic Operators....Pages 276-281 Dirac Operators on the Two-Sphere....Pages 282-288 Back Matter....Pages 289-307