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Acoustic scattering by irregular obstacles
Published in B Bertram, C Constanda, A Struthers, Integral methods in science and engineering, 2019
In 1992 the authors established asymptotics of the scattering phase s(λ) for the Dirichlet problem by different methods to Christiansen, drawing on the intimate connection between the scattering phase and the counting function N(λ) for the associated interior Dirichlet eigenvalue problem. The methods we use to discuss the Neumann problem also differ from those of Robert and Christiansen and exploit new monotonicity properties of s(λ).
The direct and inverse scattering problems for penetrable and impenetrable mixed obstacles
Published in Applicable Analysis, 2021
Let w be the unique solution of the Helmholtz equation in with boundary data , then , so we conclude that v = w in since is not a Neumann eigenvalue of in , but , so v = w = 0 in since is not a Dirichlet eigenvalue of in . Hence we derive that .
Eigenvalue estimates for the drifting Laplacian and the p-Laplacian on submanifolds of warped products
Published in Applicable Analysis, 2021
Wei Lu, Jing Mao, Chuan-Xi Wu, Ling-Zhong Zeng
It is known that (1) has a positive weak solution, which is unique modulo the scaling, in , the completion of the set of smooth functions compactly supported on under the Sobolev norm , and the first Dirichlet eigenvalue of the p-Laplacian in the BVP (1) can be characterized by where dv is the volume element of M. By Domain monotonicity of eigenvalues with vanishing Dirichlet data for the first Dirichlet eigenvalue of (see, e.g. [1, Lemma 1.1]), for a complete noncompact Riemannian manifold M and a geodesic ball , with center q and radius r, on M, one can define the following limit which is independent of the choice of the point q.
Improved integral formulae for supersonic reconstruction of the acoustic field
Published in Inverse Problems in Science and Engineering, 2018
In (7) and (8), is the surface density function. When is a Dirichlet eigenvalue of the Laplace operator in then it is known that there is no unique such that the single layer representation in (7) holds. A similar situation is found for the double layer representation in (8) when for a Neumann eigenvalue of the Laplace operator in . This situation is avoided using combined representation [13] (also known as the Burton-Miller representation [19] in the acoustic literature)