Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Rivière[1]

  • [1] Université Lille 1 U.F.R. de Mathématiques Laboratoire Paul Painlevé (U.M.R. CNRS 8524) 59655 Villeneuve d’Ascq Cedex (France)

Annales de l’institut Fourier (2014)

  • Volume: 64, Issue: 3, page 1229-1267
  • ISSN: 0373-0956

Abstract

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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 β -damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β -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.

How to cite

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Rivière, Gabriel. "Eigenmodes of the damped wave equation and small hyperbolic subsets." Annales de l’institut Fourier 64.3 (2014): 1229-1267. <http://eudml.org/doc/275553>.

@article{Rivière2014,
abstract = {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.},
affiliation = {Université Lille 1 U.F.R. de Mathématiques Laboratoire Paul Painlevé (U.M.R. CNRS 8524) 59655 Villeneuve d’Ascq Cedex (France); Institut de Physique Théorique (CEA Saclay) Orme des Cerisiers, CEA Saclay 91191 Gif-sur-Yvette Cedex (France)},
author = {Rivière, Gabriel},
journal = {Annales de l’institut Fourier},
keywords = {nonselfadjoint operators; semiclassical analysis; eigenmodes; damped wave equation; uniform hyperbolicity; topological pressure; non self adjoint operators},
language = {eng},
number = {3},
pages = {1229-1267},
publisher = {Association des Annales de l’institut Fourier},
title = {Eigenmodes of the damped wave equation and small hyperbolic subsets},
url = {http://eudml.org/doc/275553},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Rivière, Gabriel
TI - Eigenmodes of the damped wave equation and small hyperbolic subsets
JO - Annales de l’institut Fourier
PY - 2014
PB - Association des Annales de l’institut Fourier
VL - 64
IS - 3
SP - 1229
EP - 1267
AB - 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.
LA - eng
KW - nonselfadjoint operators; semiclassical analysis; eigenmodes; damped wave equation; uniform hyperbolicity; topological pressure; non self adjoint operators
UR - http://eudml.org/doc/275553
ER -

References

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