Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces

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Chandler-Wilde, Simon Neil ; Elschner, J (2010)

We consider the problem of scattering of time harmonic acoustic waves by an unbounded sound soft surface which is assumed to lie within a finite distance of some plane. The paper is concerned with the study of an equivalent variational formulation of this problem set in a scale of weighted Sobolev spaces. We prove well-posedness of this variational formulation in an energy space with weights which extends previous results in the unweighted setting [S. Chandler-Wilde and P. Monk, SIAM J. Math. Anal., 37 (2005), pp. 598–618] to more general inhomogeneous terms in the Helmholtz equation. In particular, in the two-dimensional case, our approach covers the problem of plane wave incidence, whereas in the three-dimensional case, incident spherical and cylindrical waves can be treated. As a further application of our results, we analyze a finite section type approximation, whereby the variational problem posed on an infinite layer is approximated by a variational problem on a bounded region.
  • References (43)
    43 references, page 1 of 5

    [1] H. Ammari, G. Bao, and A. W. Wood, Analysis of the electromagnetic scattering from a cavity, Japan J. Indust. Appl. Math., 19 (2002), pp. 301-310.

    [2] T. Arens, S. N. Chandler-Wilde, and K. Haseloh, Solvability and spectral properties of integral equations on the real line: I. Weighted spaces of continuous functions, J. Math. Anal. Appl., 272 (2002), pp. 276-302.

    [3] T. Arens, S. N. Chandler-Wilde, and K. Haseloh, Solvability and spectral properties of integral equations on the real line. II. Lp-spaces and applications, J. Integral Equations Appl., 15 (2003), pp. 1-35.

    [4] T. Arens and T. Hohage, On radiation conditions for rough surface scattering problems, IMA J. Appl. Math., 70 (2005), pp. 839-847.

    [5] G. Bao and D. C. Dobson, On the scattering by a biperiodic structure, Proc. Amer. Math. Soc., 128 (2000), pp. 2715-2723.

    [6] A. S. Bonnet-Bendhia and P. Starling, Guided waves by electromagnetic gratings and nonuniqueness examples for the diffraction problem, Math. Methods Appl. Sci., 17 (1994), pp. 305-338.

    [7] M. Cessenat, Mathematical Methods in Electromagnetism, World Scientific, Singapore, 1996.

    [8] S. N. Chandler-Wilde, Boundary value problems for the Helmholtz equation in a half-plane, in Third International Conference on Mathematical and Numerical Aspects of Wave Propagation (Proceedings in Applied Mathematics), G. Cohen, ed., SIAM, Philadelphia, 1995, pp. 188-197.

    [9] S. N. Chandler-Wilde and B. Zhang, Electromagnetic scattering by an inhomogeneous conducting or dielectric layer on a perfectly conducting plate, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., A454 (1998), pp. 519-542.

    [10] S. N. Chandler-Wilde and B. Zhang, A uniqueness result for scattering by infinite rough surfaces, SIAM J. Appl. Math., 58 (1998), pp. 1774-1790.

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