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Asymptotic behaviour of the phase in non-smooth obstacle scattering

Authors:

Chen Hua

Institution:

(1) Institute of Mathematics, Wuhan University, 430072 Wuhan, China

Abstract:

In this paper, we study the asymptotic behaviour of the scattering phases(λ) of the Dirichlet Laplacian associated with obstacle $\bar \Omega$ , where Ω is a bounded open subset of ℝⁿ (n≥2) with non-smooth boundary ∂Ω and connected complement Ω_e =ℝⁿ $\bar \Omega$ . We can prove that if Ω satisfies a certain geometrical condition, then

$\left| {s(\lambda ) - \phi (\lambda )} \right| \leqslant d_n |\partial \Omega |_{n - 1} \lambda ^{\frac{{n - 1}}{2}} \log \lambda , as\lambda \to + \infty$

where $\phi (\lambda ) = \left {(4\pi )^{\tfrac{n}{2}} \Gamma \left( {1 + \tfrac{n}{2}} \right)} \right]^{ - 1} \left| \Omega \right|_n \lambda ^{\tfrac{n}{2}} ,d_n > 0$ ,d _n>0 depending only onn, and |·|_j (j = n - l, n) is aj- dimensional Lebesgue measure. Research partially supported by the Natural Science Foundation of China and the Grant of Chinese State Education Committee

Keywords:

Scattering phase Counting function Dirichlet Laplacian Obstacle Exterior boundary problem Tessellation of domains

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