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 time-harmonic elastic wave scattering by one-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 at arbitrary frequency for both elastic plane wave and point source (spherical) wave incidence in the two-dimensional case. In particular, our approach covers the elastic scattering from periodic structures (diffraction gratings), and we prove quasiperiodicity of the scattered field whenever the incident field is quasiperiodic. Moreover, the diffraction grating problem is also uniquely solvable in the presented weighted Sobolev spaces for a broad class of non-quasiperiodic incident waves.

Keywords:

• non-smooth rough surface
• linear elasticity
• radiation condition
• variational formulation
• weighted Sobolev spaces
• Navier equation

AMS Subject Classifications:

• 74B05
• 35J05
• 35J20
• 35J25
• 42B10
• 78A45
• 74J20
• 35J57
• 35Q74

Acknowledgement

The second author gratefully acknowledges the support by the German Research Foundation (DFG) under Grant No. HU 2111/1-1.