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

Laurent Bourgeois 1 Lucas Chesnel 2, 3 Sonia Fliss 1
1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
3 DeFI - Shape reconstruction and identification
CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique, Inria Saclay - Ile de France
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.
Type de document :
Pré-publication, Document de travail
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Contributeur : Lucas Chesnel <>
Soumis le : vendredi 28 septembre 2018 - 12:16:36
Dernière modification le : mardi 5 février 2019 - 15:28:19
Document(s) archivé(s) le : samedi 29 décembre 2018 - 13:52:54


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  • HAL Id : hal-01883560, version 1


Laurent Bourgeois, Lucas Chesnel, Sonia Fliss. On well-posedness of time-harmonic problems in an unbounded strip for a thin plate model. 2018. 〈hal-01883560〉



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