Geometric Wave Equations
Abstract
In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed including a detailed treatment of the Cauchy problem on a globally hyperbolic manifold both for the smooth and finite order setting. As application, the classical Poisson algebra of polynomial functions on the initial values and the dynamical Poisson algebra coming from the wave equation are related. The text contains an introduction to the theory of distributions on manifolds as well as detailed proofs.
 Publication:

arXiv eprints
 Pub Date:
 August 2012
 DOI:
 10.48550/arXiv.1208.4706
 arXiv:
 arXiv:1208.4706
 Bibcode:
 2012arXiv1208.4706W
 Keywords:

 Mathematics  Differential Geometry;
 Mathematical Physics;
 Mathematics  Analysis of PDEs
 EPrint:
 Updated lecture notes for a lecture held in Freiburg 2008/2009, 279 pages, many figures