Communications in Mathematical Sciences

Volume 17 (2019)

Number 6

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

Pages: 1487 – 1529

DOI: https://dx.doi.org/10.4310/CMS.2019.v17.n6.a2

Authors

Laurent Bourgeois (Laboratoire Poems, CNRS, INRIA, Ensta Paris, Institut Polytechnique de Paris, Palaiseau, France)

Lucas Chesnel (INRIA / Centre de mathématiques appliquées, École Polytechnique, Institut Polytechnique de Paris, Palaiseau, France)

Sonia Fliss (Laboratoire Poems, CNRS, INRIA, Ensta Paris, Institut Polytechnique de Paris, Palaiseau, France)

Abstract

We study the propagation of elastic waves in the time-harmonic regime in a wave-guide 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.

Keywords

waveguide, Kirchhoff–Love model, thin plate, radiation conditions, modal decomposition

2010 Mathematics Subject Classification

35B40, 35Q74, 74B15, 74H20, 74J20

Full Text (PDF format)

Received 6 December 2018

Accepted 1 July 2019

Published 26 December 2019