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Initial Value Problems for Wave Equations on Manifolds

  • Christian BärEmail author
  • Roger Tagne Wafo
Article

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.

Keywords

Wave equation Globally hyperbolic Lorentz manifold Cauchy problem Goursat problem Finite energy sections 

Mathematics Subject Classification (2010)

35L05 35L15 58J45 

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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Institut für MathematikUniversität PotsdamPotsdamGermany
  2. 2.Faculty of Science, Department of Mathematics and Computer ScienceUniversity of DoualaDoualaCameroon

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