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Singularities of the scattering kernel for generic obstacles. (English) Zbl 0729.35099

Let \(\Omega \subset {\mathbb{R}}^ n\), \(n\geq 3\), n odd, be an open connected domain with \(C^{\infty}\) smooth boundary and consider the scattering kernel s(t,\(\theta\),\(\omega\)) related to the wave equation in \({\mathbb{R}}\times \Omega\) with Dirichlet boundary conditions on \({\mathbb{R}}\times \partial \Omega\). Here \((\theta,\omega)\in S^{n-1}\times S^{n-1}.\)
The aim of this paper is to study the singular support of S(t,\(\theta\),\(\omega\)) for general (non-convex) obstacles. If \(L_{\omega,\theta}\) denotes the set of all (\(\theta\),\(\omega\))-rays, \(\gamma\) denotes a reflected ray in \({\bar \Omega}\) and \(T_{\gamma}\) denotes the sojourn time of \(\gamma\) then the authors prove:
Let \(\theta\neq \omega\) be fixed. Assume that every (\(\omega\),\(\theta\))- ray in \({\bar \Omega}\) is uniquely extendible. Then \[ \sin g \sup p S(t,\theta,\omega)\quad \subset \quad \{-T_{\gamma}:\;\gamma \in L_{\omega,\theta}\}. \] If one has equality in the above result, the authors go on to study the singular form of S.

MSC:

35P25 Scattering theory for PDEs
35L05 Wave equation
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References:

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