The geometrical quantity in damped wave equations on a square

Hébrard, Pascal; Humbert, Emmanuel

doi:10.1051/cocv:2006015

Hébrard, Pascal ; Humbert, Emmanuel

ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 636-661.

Résumé

The energy in a square membrane $Ω$ subject to constant viscous damping on a subset $ω \subset Ω$ 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.

MR Zbl

DOI : 10.1051/cocv:2006015

Classification : 35L05, 93D15
Mots clés : damped wave equation, mathematical billards

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Hébrard, Pascal; Humbert, Emmanuel. The geometrical quantity in damped wave equations on a square. ESAIM: Control, Optimisation and Calculus of Variations, Tome 12 (2006) no. 4, pp. 636-661. doi : 10.1051/cocv:2006015. http://www.numdam.org/articles/10.1051/cocv:2006015/

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