A hitchhiker's guide to first-order elliptic boundary value problems

Abstract

To empower the mathematical hitchhiker wishing to use operator methods in geometry and topology, we present this user's guide to first-order elliptic boundary value problems. Existence, regularity, and Fredholmness are discussed for general first-order elliptic operators on manifolds with compact boundary. The focus is on a very general class of elliptic boundary conditions, which contain those that are pseudo-local as a special case, yielding the relative index theorem. A new characterisation of a subclass of elliptic boundary conditions is also given.