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In this paper we describe the propagation of ${\mathcal C}^{\infty}$ and Sobolev singularities for the wave equation on ${\mathcal C}^{\infty}$ manifolds with corners $M$ equipped with a Riemannian metric $g$. That is, for $X=M\times\mathbb{R}_t$, $P=D_t^2-\Delta_M$, and $u\in H^1_{\mathrm{loc}}(X)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that $\mathrm{WF}_{b}(u)$ is a union of maximally extended generalized broken bicharacteristics. This result is a ${\mathcal{C}}^{\infty}$ 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).
@article{10_4007_annals_2008_168_749,
author = {Andr\'as Vasy},
title = {Propagation of singularities for the wave equation on manifolds with corners},
journal = {Annals of mathematics},
pages = {749--812},
publisher = {mathdoc},
volume = {168},
number = {3},
year = {2008},
doi = {10.4007/annals.2008.168.749},
mrnumber = {2456883},
zbl = {1171.58007},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.749/}
}
TY - JOUR AU - András Vasy TI - Propagation of singularities for the wave equation on manifolds with corners JO - Annals of mathematics PY - 2008 SP - 749 EP - 812 VL - 168 IS - 3 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.749/ DO - 10.4007/annals.2008.168.749 LA - en ID - 10_4007_annals_2008_168_749 ER -
%0 Journal Article %A András Vasy %T Propagation of singularities for the wave equation on manifolds with corners %J Annals of mathematics %D 2008 %P 749-812 %V 168 %N 3 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4007/annals.2008.168.749/ %R 10.4007/annals.2008.168.749 %G en %F 10_4007_annals_2008_168_749
András Vasy. Propagation of singularities for the wave equation on manifolds with corners. Annals of mathematics, Tome 168 (2008) no. 3, pp. 749-812. doi: 10.4007/annals.2008.168.749
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