The geometrical quantity in damped wave equations on a square
| dc.creator | Hébrard, Pascal | |
| dc.creator | Humbert, Emmanuel | |
| dc.date | 2007-06-01 | |
| dc.date.accessioned | 2026-07-07T08:03:54Z | |
| dc.date.available | 2026-07-07T08:03:54Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/0706.0172 | |
| dc.identifier | http://arxiv.org/abs/0706.0172 | |
| dc.identifier | ESAIM - Control Optimisation and Calculs of Variations 12, 4 (31/12/2006) 636-661 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/129750 | |
| dc.subject | Differential Geometry | |
| dc.subject | 35L05, 93D15 | |
| dc.title | The geometrical quantity in damped wave equations on a square | |
| dc.type | text |