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 ω ⊂ Ω decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate τ ( ω ) of this decay satisfies τ ( ω ) = 2 min ( - μ ( ω ) , g ( ω ) ) (see Lebeau [Math. Phys. Stud.19 (1996) 73–109]). Here μ ( ω ) denotes the spectral abscissa of the damped wave equation operator and  g ( ω ) 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 ( ω ) is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly g ( ω ) 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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