Diffraction of singularities for the wave equation on manifolds with corners
type: Item
instance of: scholarly article
zbMATH Open document ID: 1277.35004
author: Richard Melrose
author: András Vasy
author: Jared Wunsch
publication date: 2013-07-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/0903.3208
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35A18
Mathematics Subject Classification ID: 35A21
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L67
Mathematics Subject Classification ID: 58J47
Mathematics Subject Classification ID: 58J45
zbMATH DE Number: 6189962
zbMATH Keywords: wave equation
zbMATH Keywords: manifold with corners
zbMATH Keywords: propagation of singularities
zbMATH Keywords: singularity reflection from corners
MaRDI profile type: Publication
arXiv ID: 0903.3208
arXiv classification: math.AP
arXiv classification: math-ph
arXiv classification: math.MP
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title: Diffraction of singularities for the wave equation on manifolds with corners
published in: Astérisque
review text: This Astérisque volume is devoted to the study of the propagation of the singularities for the solutions of the wave equation NEWLINE\[NEWLINE\square u= D^2_t u-\Delta u= 0,NEWLINE\]NEWLINE where \(\Delta\) is the Laplace-Beltrami operator on a manifold with corners \(M\). The behaviour of the microlocal singularities in the interior of \(M\times\mathbb{R}\) is well understood, see \textit{J. J. Duistermaat} and \textit{L. Hörmander} [Acta Math. 128, 183--269 (1972; Zbl 0232.47055)], namely they propagate along the null-bicharacteristics. Moreover it is known from \textit{J. Chazarain} [Publ. Res. Inst. Math. Sci., Kyoto Univ. 12, Suppl., 39--52 (1977; Zbl 0365.35050)] that singularities striking the boundary of \(M\) transversely, at a smooth point, simple reflect. Attention is here fixed on the case when the boundary of \(M\) is not smooth and the wave front sets intersect with a corner. This gives rise to new different phenomena, namely the singularity reflects from the corner along a whole family of bicharacteristics. The authors give a detailed analysis of these phenomena, collecting and improving their preceding results on the subject. The basis technique is to the blow up the corner, and define in this setting geometric broken characteristics passing through the corner. With respect to preceding contributions of the authors, here particular attention is paid to the Sobolev regularity of the diffracted wave front set, which can be weaker than the original regularity. The volume contains some nice figures, which explain the geometric setting, and help the intuition.
reviewed by: Luigi Rodino
description: scientific article; zbMATH DE number 6189962