Propagation of singularities for the wave equation on manifolds with corners

Abstract

In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Riemannian metric g.That is, for X = M × R t , P = D 2 t -∆ M , and u ∈ H 1 loc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics.This result is a C ∞ counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11].Our methods rely on b-microlocal positive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners).

Locations

• arXiv (Cornell University) - View - PDF
• French digital mathematics library (Numdam) - View - PDF
• Annals of Mathematics - View - PDF