Propagation of singularities for the wave equation on manifolds with corners

2004/05/22 by András Vasy, Andras Vasy, Vasy, Andras · 3 citations
Mathematics · #58J47 (Primary) 35L20 (Secondary) #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics #math.AP #msc:35L20 #msc:58J47

paper · pdf · doi:10.48550/arxiv.math/0405431

arxiv created 2004/05/22 · openalex publication_date 2004/05/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we describe the propagation of smooth (C^∞) and Sobolev singularities for the wave equation on smooth manifolds with corners M equipped with a Riemannian metric g. That is, for X=MxR, P=D_t²-Δ_M, and u locally in H¹ solving Pu=0 with homogeneous Dirichlet or Neumann boundary conditions, we show that the wave front set of u is a union of maximally extended generalized broken bicharacteristics. This result is a smooth counterpart of Lebeau's results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary. 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).

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• The Dirichlet-to-Neumann map on asymptotically anti-de Sitter spaces and holography
• Propagation of singularities for equations with C^r coefficients for r>1
• Singularities of Dirac-Coulomb propagators

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