Boundary Value Problems for Elliptic Differential Operators of First Order

TL;DR

This paper develops a comprehensive theory for boundary value problems of first-order elliptic operators, especially Dirac type, including boundary conditions, regularity, and index theory, generalizing classical results and proving new theorems.

Contribution

It provides a unified, simplified framework for elliptic boundary value problems of Dirac type operators, extending classical theories and introducing new decomposition and index theorems.

Findings

Generalized elliptic boundary conditions for Dirac operators

Proved a decomposition theorem for boundary value problems

Extended Gromov-Lawson's relative index theorem and cobordism theorem

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 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.