Propagation of singularities for the wave equation on manifolds with corners

András Vasy[1]

  • [1] Department of Mathematics, MIT and Northwestern University

Séminaire Équations aux dérivées partielles (2004-2005)

  • Volume: 2004-2005, page 1-16

Abstract

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In this talk 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 t 2 - Δ M , and u ∈ H loc 1 ( 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, [7]. 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).These notes are a summary of [17], where the detailed proofs appear.

How to cite

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Vasy, András. "Propagation of singularities for the wave equation on manifolds with corners." Séminaire Équations aux dérivées partielles 2004-2005 (2004-2005): 1-16. <http://eudml.org/doc/11110>.

@article{Vasy2004-2005,
abstract = {In this talk 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 $u\in H^1_\{\{\text\{loc\}\}\}(X)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that $\operatorname\{WF\}_\{\{\text\{b\}\}\}(u)$ 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, [7]. 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).These notes are a summary of [17], where the detailed proofs appear.},
affiliation = {Department of Mathematics, MIT and Northwestern University},
author = {Vasy, András},
journal = {Séminaire Équations aux dérivées partielles},
language = {fre},
pages = {1-16},
publisher = {Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {Propagation of singularities for the wave equation on manifolds with corners},
url = {http://eudml.org/doc/11110},
volume = {2004-2005},
year = {2004-2005},
}

TY - JOUR
AU - Vasy, András
TI - Propagation of singularities for the wave equation on manifolds with corners
JO - Séminaire Équations aux dérivées partielles
PY - 2004-2005
PB - Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2004-2005
SP - 1
EP - 16
AB - In this talk 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 $u\in H^1_{{\text{loc}}}(X)$ solving $Pu=0$ with homogeneous Dirichlet or Neumann boundary conditions, we show that $\operatorname{WF}_{{\text{b}}}(u)$ 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, [7]. 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).These notes are a summary of [17], where the detailed proofs appear.
LA - fre
UR - http://eudml.org/doc/11110
ER -

References

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  14. M. Mitrea, M. Taylor, and A. Vasy. Lipschitz domains, domains with corners and the Hodge Laplacian. Preprint, 2004. Zbl1082.31007MR2182300
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  17. A. Vasy. Propagation of singularities for the wave equation on manifolds with corners. Preprint, 2004. Zbl1266.58013

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