ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - October 2006 - Volume 12, Issue 04  

Research Article

The geometrical quantity in damped wave equations on a square

Hébrard, Pascala1 and Humbert, Emmanuela1

a1 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

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.

(Received November 14 2003)

(Revised July 19 2004)

(Revised June 13 2005)

(Online publication October 11 2006)

Key Words:

  • Damped wave equation;
  • mathematical billards.

Mathematics Subject Classification:

  • 35L05;
  • 93D15