Elastic scattering by unbounded rough surfaces : solvability in weighted Sobolev spaces

Abstract

This paper is concerned with the variational approach in weighted Sobolev spaces to timeharmonic elastic scattering by two-dimensional unbounded rough surfaces. The rough surface is supposed to be the graph of a bounded and uniformly Lipschitz continuous function, on which the total elastic displacement satisfies either the Dirichlet or impedance boundary condition. We establish uniqueness and existence results for both elastic plane and point source (spherical) wave incidence, following the recently developed variational approach in [SIAM J. Math. Anal., 42: 6 (2010), pp. 2554 2580] for the Helmholtz equation. This paper extends our previous solvability results [SIAM J. Math. Anal., 44: 6 (2012), pp. 4101-4127] in the standard Sobolev space to the weighted Sobolev spaces.