Initial value problems for wave equations on manifolds

Christian Baer, Roger Tagne Wafo

Published 2014-08-21Version 1

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. We also show existence and uniqueness of solutions for the Goursat problem where one prescribes initial data on a characteristic Cauchy hypersurface. This extends classical results due to H\"ormander.