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 -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of -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.
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 -amorties ne peut pas être complètement concentrée dans des petits voisinages d’un petit sous-ensemble hyperbolique fixé qui est formé de trajectoires -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.
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
Rivière, Gabriel 1
@article{AIF_2014__64_3_1229_0, author = {Rivi\`ere, Gabriel}, title = {Eigenmodes of the damped wave equation and small hyperbolic subsets}, journal = {Annales de l'Institut Fourier}, pages = {1229--1267}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {64}, number = {3}, year = {2014}, doi = {10.5802/aif.2879}, mrnumber = {3330169}, zbl = {06387306}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2879/} }
TY - JOUR AU - Rivière, Gabriel TI - Eigenmodes of the damped wave equation and small hyperbolic subsets JO - Annales de l'Institut Fourier PY - 2014 SP - 1229 EP - 1267 VL - 64 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2879/ DO - 10.5802/aif.2879 LA - en ID - AIF_2014__64_3_1229_0 ER -
%0 Journal Article %A Rivière, Gabriel %T Eigenmodes of the damped wave equation and small hyperbolic subsets %J Annales de l'Institut Fourier %D 2014 %P 1229-1267 %V 64 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2879/ %R 10.5802/aif.2879 %G en %F AIF_2014__64_3_1229_0
Rivière, Gabriel. Eigenmodes of the damped wave equation and small hyperbolic subsets. Annales de l'Institut Fourier, Volume 64 (2014) no. 3, pp. 1229-1267. doi : 10.5802/aif.2879. https://aif.centre-mersenne.org/articles/10.5802/aif.2879/
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