Geometric optics expansions with amplification for hyperbolic boundary value problems: Linear problems

Coulombel, Jean-François; Guès, Olivier

Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2183-2233.

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.

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.

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