The geometrical quantity in damped wave equations on a square

Pascal Hébrard; Emmanuel Humbert

ESAIM: Control, Optimisation and Calculus of Variations (2006)

  • Volume: 12, Issue: 4, page 636-661
  • ISSN: 1292-8119

Abstract

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The energy in a square membrane Ω subject to constant viscous damping on a subset decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate of this decay satisfies (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here denotes the spectral abscissa of the damped wave equation operator and  is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion of a mass-point in Ω subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly when ω is a finite union of squares.

How to cite

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Hébrard, Pascal, and Humbert, Emmanuel. "The geometrical quantity in damped wave equations on a square." ESAIM: Control, Optimisation and Calculus of Variations 12.4 (2006): 636-661. <http://eudml.org/doc/249667>.

@article{Hébrard2006,
abstract = { The energy in a square membrane Ω subject to constant viscous damping on a subset $\omega\subset \Omega$ decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min( -\mu(\omega), g(\omega))$ (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here $\mu(\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\omega)$ is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion of a mass-point in Ω subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when ω is a finite union of squares. },
author = {Hébrard, Pascal, Humbert, Emmanuel},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Damped wave equation; mathematical billards.; mathematical billards; uniform stabilization},
language = {eng},
month = {10},
number = {4},
pages = {636-661},
publisher = {EDP Sciences},
title = {The geometrical quantity in damped wave equations on a square},
url = {http://eudml.org/doc/249667},
volume = {12},
year = {2006},
}

TY - JOUR
AU - Hébrard, Pascal
AU - Humbert, Emmanuel
TI - The geometrical quantity in damped wave equations on a square
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2006/10//
PB - EDP Sciences
VL - 12
IS - 4
SP - 636
EP - 661
AB - The energy in a square membrane Ω subject to constant viscous damping on a subset $\omega\subset \Omega$ decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau(\omega)$ of this decay satisfies $\tau(\omega)= 2 \min( -\mu(\omega), g(\omega))$ (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here $\mu(\omega)$ denotes the spectral abscissa of the damped wave equation operator and $g(\omega)$ is a number called the geometrical quantity of ω and defined as follows. A ray in Ω is the trajectory generated by the free motion of a mass-point in Ω subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g(\omega)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g(\omega)$ when ω is a finite union of squares.
LA - eng
KW - Damped wave equation; mathematical billards.; mathematical billards; uniform stabilization
UR - http://eudml.org/doc/249667
ER -

References

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