Diffraction of singularities for the wave equation on manifolds with corners
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.
scientific article; zbMATH DE number 6189962
scientific article