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

Chandler-Wilde, S. N. ORCID: https://orcid.org/0000-0003-0578-1283 and Elschner, J. (2010) Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces. SIAM Journal on Mathematical Analysis (SIMA), 42 (6). pp. 2554-2580. ISSN 0036-1410 doi: 10.1137/090776111

Abstract/Summary

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.

Item Type	Article
URI	https://reading-clone.eprints-hosting.org/id/eprint/18779
Identification Number/DOI	10.1137/090776111
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Uncontrolled Keywords	nonsmooth boundary, radiation condition, variational formulation, weighted Sobolev spaces, Helmholtz equation
Publisher	Society for Industrial and Applied Mathematics
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