Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2183-2233.

We compute and justify rigorous geometric optics expansions for linear hyperbolic boundary value problems that do not satisfy the uniform Lopatinskii condition. We exhibit an amplification phenomenon for the reflection of small high frequency oscillations at the boundary. Our analysis has two important consequences for such hyperbolic boundary value problems. Firstly, we make precise the optimal energy estimate in Sobolev spaces showing that losses of derivatives must occur from the source terms to the solution. Secondly, we are able to derive a lower bound for the finite speed of propagation, showing that waves may propagate faster than for the propagation in free space. We illustrate our analysis with some examples.

Nous calculons et justifions rigoureusement des développements d’optique géométrique pour des problèmes aux limites hyperboliques ne satisfaisant pas la condition de Lopatinskii uniforme. Nous mettons en évidence un phénomène d’amplification pour la réflexion au bord d’oscillations haute fréquence et de petite amplitude. Notre analyse induit deux conséquences importantes pour de tels problèmes aux limites. Tout d’abord, nous précisons la perte de régularité optimale dans l’échelle des espaces de Sobolev entre les termes source et la solution du problème. Ensuite, nous donnons une borne inférieure pour la vitesse finie de propagation, celle-ci pouvant être supérieure à la vitesse de propagation libre dans tout l’espace. Nous illustrons notre analyse par quelques exemples.

DOI: 10.5802/aif.2581
Classification: 35L50,  78A05
Keywords: Hyperbolic systems, boundary value problems, geometric optics
Coulombel, Jean-François 1; Guès, Olivier 2

1 CNRS & Université Lille 1 Laboratoire Paul Painlevé (UMR CNRS 8524) and Project Team SIMPAF of INRIA Lille Nord Europe Cité scientifique, Bâtiment M2 59655 VILLENEUVE D’ASCQ Cedex (France)
2 Université de Provence Laboratoire d’Analyse, Topologie et Probabilités (UMR CNRS 6632) Technopôle Château-Gombert 39 rue F. Joliot Curie 13453 MARSEILLE Cedex 13 (France)
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Coulombel, Jean-François; Guès, Olivier. Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2183-2233. doi : 10.5802/aif.2581. http://www.numdam.org/articles/10.5802/aif.2581/

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