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Asymptotic behaviour of the phase in non-smooth obstacle scattering. (English) Zbl 0880.35089

Summary: We study the asymptotic behaviour of the scattering phase \(s(\lambda)\) of the Dirichlet-Laplacian associated with obstacle \(\overline\Omega\), where \(\Omega\) is a bounded open subset of \(\mathbb{R}^n\) \((n\geq 2)\) with non-smooth boundary \(\partial\Omega\) and connected complement \(\Omega_\varepsilon= \mathbb{R}^n\backslash\overline \Omega\). We prove that if \(\Omega\) satisfies a certain geometrical condition, then \[ |s(\lambda)- \phi(\lambda)|\leq d_n|\partial\Omega|_{n- 1}\lambda^{{n-1\over 2}}\log\lambda,\quad\text{as }\lambda\to+\infty, \] where \(\phi(\lambda)= \left[(4\pi)^{n/2}\Gamma\left(1+ {n\over 2}\right)\right]^{-1}|\Omega|_n\lambda^{n/2}\), \(d_n>0\) depending only on \(n\), and \(|\cdot|_j\) \((j=n- 1,n)\) is a \(j\)-dimensional Lebesgue measure.

MSC:

35P25 Scattering theory for PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35P20 Asymptotic distributions of eigenvalues in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
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