Variational approach in weighted Sobolev spaces to scattering by unbounded rough surfaces
Chandler-Wilde, S. N. 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
To link to this article DOI: 10.1137/090776111
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.