## 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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