TAILIEUCHUNG - Đề tài " Propagation of singularities for the wave equation on manifolds with corners "

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 Rie2 1 mannian metric g. That is, for X = M × Rt , P = Dt − ∆M , and u ∈ Hloc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary conditions, we show that WFb (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. | Annals of Mathematics Propagation of singularities for the wave equation on manifolds with corners By Andr_as Vasy Annals of Mathematics 168 2008 749-812 Propagation of singularities for the wave equation on manifolds with corners By Andras Vasy Abstract In this paper we describe the propagation of C1 and Sobolev singularities for the wave equation on C1 manifolds with corners M equipped with a Rie-mannian metric g. That is for X M X Rt P D2 Xm and u 2 H0C X solving Pu 0 with homogeneous Dirichlet or Neumann boundary conditions we show that WFb u is a union of maximally extended generalized broken bicharacteristics. This result is a C1 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 . 1. Introduction In this paper we describe the propagation of C1 and Sobolev singularities for the wave equation on a manifold with corners M equipped with a smooth Riemannian metric g. We first recall the basic definitions from 12 and refer to 20 2 as a more accessible reference. Thus a tied or t- manifold with corners X of dimension n is a paracompact Hausdorff topological space with a C1 structure with corners. The latter simply means that the local coordinate charts map into 0 i k X Rn-k rather than into Rn. Here k varies with the coordinate chart. We write d X for the set of points p 2 X such that in any local coordinates ộ ộ1 ộk ộk 1 ộn near p with k as above precisely of the first k coordinate functions vanish at ộ p . We usually write such local coordinates as x1 xk y1 yn-k . A boundary face of codimension is the closure of a connected component of @ X. A boundary face of codimension 1 is called a boundary hypersurface. A manifold with corners is a tied manifold with corners such that .

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