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Guide to Boundary Value Problems for Dirac-Type Operators

Christian Baer, Werner Ballmann

Abstract

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

Guide to Boundary Value Problems for Dirac-Type Operators

Abstract

We present an introduction to boundary value problems for Dirac-type operators on complete Riemannian manifolds with compact boundary. We introduce a very general class of boundary conditions which contains local elliptic boundary conditions in the sense of Lopatinskij and Shapiro as well as the Atiyah-Patodi-Singer boundary conditions. We discuss boundary regularity of solutions and also spectral and index theory. The emphasis is on providing the reader with a working knowledge.

Paper Structure

This paper contains 26 sections, 30 theorems, 152 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Differential operators
  4. The principal symbol
  5. Elliptic operators
  6. Dirac-type operators
  7. Clifford relations and Dirac-type operators
  8. Adapted operators on the boundary
  9. Formally self-adjoint Dirac-type operators
  10. Boundary value problems
  11. Spectral subspaces
  12. The maximal domain
  13. Boundary conditions
  14. D-elliptic boundary conditions
  15. Self-adjoint D-elliptic boundary conditions
  16. ...and 11 more sections

Key Result

Proposition 2.1

Let $D$ be a differential operator from $E$ to $F$ of order one. Then we have, for all $\Phi\in C^\infty_c(M,E)$ and $\Psi\in C^\infty_c(M,F)$,

Figures (2)

  • Figure 1: Cutting $M$ along the hypersurface $N$
  • Figure 2: Geodesic ball of radius $r$ in the unit sphere

Theorems & Definitions (67)

  • Proposition 2.1: Green's formula
  • Proposition 2.3
  • Corollary 2.4
  • Proposition 3.1: Weitzenböck formula
  • Lemma 3.2
  • Theorem 4.2
  • Remark 4.3
  • Theorem 4.4: Boundary regularity I, BB
  • Definition 4.5
  • Theorem 4.6: The adjoint operator, BB
  • ...and 57 more