The geometrical quantity in damped wave equations on a square

Hébrard, Pascal; Humbert, Emmanuel

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

The energy in a square membrane $\Omega$ subject to constant viscous damping on a subset $\omega \subset \Omega$ decays exponentially in time as soon as $\omega$ satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate $\tau \left(\omega \right)$ of this decay satisfies $\tau \left(\omega \right)=2min(-\mu (\omega ),g(\omega \left)\right)$ (see Lebeau [Math. Phys. Stud. 19 (1996) 73-109]). Here $\mu \left(\omega \right)$ denotes the spectral abscissa of the damped wave equation operator and $g\left(\omega \right)$ is a number called the geometrical quantity of $\omega$ and defined as follows. A ray in $\Omega$ is the trajectory generated by the free motion of a mass-point in $\Omega$ subject to elastic reflections on the boundary. These reflections obey the law of geometrical optics. The geometrical quantity $g\left(\omega \right)$ is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly $g\left(\omega \right)$ when $\omega$ is a finite union of squares.

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

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