Eigenmodes of the damped wave equation and small hyperbolic subsets

Rivière, Gabriel

Annales de l'Institut Fourier, Tome 64 (2014) no. 3, pp. 1229-1267.

Sur une variété riemannienne, lisse, compacte et sans bord, on étudie les solutions stationnaires de l’équation des ondes amorties. Dans la limite haute fréquence, on démontre qu’une suite de solutions stationnaires $\beta$-amorties ne peut pas être complètement concentrée dans des petits voisinages d’un petit sous-ensemble hyperbolique fixé qui est formé de trajectoires $\beta$-amorties du flot géodésique.

L’article contient aussi un appendice (de S. Nonnenmacher et de l’auteur) dans lequel on établit l’existence d’une bande de taille inverse logarithmique sans valeurs propres en dessous de l’axe réel lorsque l’ensemble des trajectoires non amorties vérifie une hypothèse de pression négative.

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of $\beta$-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of $\beta$-damped trajectories of the geodesic flow.

The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.

Classification : 58J51, 35P20, 81Q12, 81Q20, 37D20
Keywords: nonselfadjoint operators, semiclassical analysis, eigenmodes, damped wave equation, uniform hyperbolicity, topological pressure
Mot clés : opérateurs non auto-adjoints, analyse semi-classique, modes propres, équation des ondes amorties, hyperbolicité uniforme, pression topologique

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