The geometrical quantity in damped wave equations on a square

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

Received: 14 November 2003
Revised: 19 July 2004
Revised: 13 June 2005

Abstract

The energy in a square membrane Ω subject to constant viscous damping on a subset decays exponentially in time as soon as ω satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate of this decay satisfies (see Lebeau [Math. Phys. Stud. 19 (1996) 73–109]). Here denotes the spectral abscissa of the damped wave equation operator and  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 is then defined as the upper limit (large time asymptotics) of the average trajectory length. We give here an algorithm to compute explicitly when ω is a finite union of squares.