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Issue ESAIM: COCV
Volume 12, Number 4, October 2006
Page(s) 636 - 661
DOI 10.1051/cocv:2006015
Published online 11 October 2006

ESAIM: COCV, October 2006, Vol. 12, pp. 636-661
DOI: 10.1051/cocv:2006015

The geometrical quantity in damped wave equations on a square

Pascal Hébrard and Emmanuel Humbert

Institut Élie Cartan, Université de Nancy 1, BP 239 54506 Vandoeuvre-lès-Nancy Cedex, France; pascal_hebrard@ds-fr.com; humbert@iecn.u-nancy.fr


(Received November 14, 2003. Revised July 19, 2004 and June 13, 2005. Published online 11 October 2006.)

Abstract
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 $\tau(\omega)$ 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 $\omega$ 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 $g(\omega)$ 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 $\omega$ is a finite union of squares.


Mathematics Subject Classification. 35L05, 93D15

Key words: Damped wave equation, mathematical billards.


© EDP Sciences, SMAI 2006


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