zbMATH — the first resource for mathematics

Boundary value problems for elliptic differential operators of first order. (English) Zbl 1331.58022
Cao, Huai-Dong (ed.) et al., In memory of C. C. Hsiung. Lectures given at the JDG symposium on geometry and topology, Lehigh University, Bethlehem, PA, USA, May 28–30, 2010. Somerville, MA: International Press (ISBN 978-1-57146-237-4/hbk). Surveys in Differential Geometry 17, 1-78 (2012).
Summary: 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 selfcontained 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.
For the entire collection see [Zbl 1239.00059].

58J05 Elliptic equations on manifolds, general theory
58J20 Index theory and related fixed-point theorems on manifolds
35J56 Boundary value problems for first-order elliptic systems
58J32 Boundary value problems on manifolds
Full Text: arXiv