Surveys in Differential Geometry

Volume 17 (2012)

Boundary value problems for elliptic differential operators of first order

Pages: 1 – 78

DOI: http://dx.doi.org/10.4310/SDG.2012.v17.n1.a1

Authors

Werner Ballmann (Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany.)

Christian Bär (Universität Potsdam, Institut für Mathematik, Am Neuen Palais 10, Haus 8, 14469 Potsdam, Germany.)

Abstract

We study boundary value problems for linear elliptic differential operators of order one. The underlying manifold may be noncompact, but the boundary is assumed to be compact. We require a symmetry property of the principal symbol of the operator along the boundary. This is satisfied by Dirac type operators, for instance.

We provide a self contained introduction to (nonlocal) elliptic boundary conditions, boundary regularity of solutions, and index theory. In particular, we simplify and generalize the traditional theory of elliptic boundary value problems for Dirac type operators. We also prove a related decomposition theorem, a general version of Gromov and Lawson’s relative index theorem and a generalization of the cobordism theorem.

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