# 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

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topHébrard, 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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- G. Lebeau, Équation des ondes amorties, in Algebraic and Geometric Methods in Mathematical Physics. Kluwer Acad. Publ., Math. Phys. Stud.19 (1996) 73–109.
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