Initial value problems for wave equations on manifolds

TL;DR

This paper develops a comprehensive theory for linear wave equations on Lorentz manifolds, establishing well-posedness, invariance of solution spaces, and extending classical results to Goursat problems.

Contribution

It introduces invariant finite energy solution spaces and proves existence and uniqueness for Goursat problems on Lorentz manifolds.

Findings

Well-posedness of the Cauchy problem in finite energy spaces

Invariance of certain solution spaces under choice of time function

Existence and uniqueness results for Goursat problems

Abstract

We study the global theory of linear wave equations for sections of vector bundles over globally hyperbolic Lorentz manifolds. We introduce spaces of finite energy sections and show well-posedness of the Cauchy problem in those spaces. These spaces depend in general on the choice of a time function but it turns out that certain spaces of finite energy solutions are independent of this choice and hence invariantly defined. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic partial Cauchy hypersurface. This extends classical results due to H\"ormander.