Initial value problems for wave equations on manifolds

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 Hörmander.

weakened assumptions in the theorem on the Goursat problem, references added, some more minor modifications

References in corpus (4)

• Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
• Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• Black hole formation from a complete regular past

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