On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model

TL;DR

This paper establishes the well-posedness of time-harmonic elastic wave propagation in a thin, unbounded waveguide modeled as a Kirchhoff-Love plate, using different boundary conditions and advanced mathematical techniques.

Contribution

It develops a rigorous framework for the well-posedness of scattering problems in a perturbed 2D strip with thin plate assumptions, handling both simply supported and clamped boundary conditions.

Findings

Well-posedness is proven using Hilbert basis for simply supported cases.

Kondratiev's approach with Fourier transform is used for clamped boundary conditions.

Solutions satisfy the outgoing radiation condition and are physically meaningful.

Abstract

We study the propagation of elastic waves in the time-harmonic regime in a waveguide which is unbounded in one direction and bounded in the two other (transverse) directions. We assume that the waveguide is thin in one of these transverse directions, which leads us to consider a Kirchhoff-Love plate model in a locally perturbed 2D strip. For time harmonic scattering problems in unbounded domains, well-posedness does not hold in a classical setting and it is necessary to prescribe the behaviour of the solution at infinity. This is challenging for the model that we consider and constitutes our main contribution. Two types of boundary conditions are considered: either the strip is simply supported or the strip is clamped. The two boundary conditions are treated with two different methods. For the simply supported problem, the analysis is based on a result of Hilbert basis in the transverse section. For the clamped problem, this property does not hold. Instead we adopt the Kondratiev's approach, based on the use of the Fourier transform in the unbounded direction, together with techniques of weighted Sobolev spaces with detached asymptotics. After introducing radiation conditions, the corresponding scattering problems are shown to be well-posed in the Fredholm sense. We also show that the solutions are the physical (outgoing) solutions in the sense of the limiting absorption principle.