标题： 具有阻尼波方程的方形上的几何量 显示英文标题

标题： The geometrical quantity in damped wave equations on a square

摘要： 一个正方形膜的能量 $\Omega$，当其在一个子集 $\omega\subset \Omega$上受到恒定粘性阻尼时，只要 $\omega$满足一个被称为“Bardos-Lebeau-Rauch”条件的几何条件，就会随时间以指数方式衰减。 衰减率 $\tau(\omega)$满足 $\tau(\omega)= 2 \min(-\mu(\omega), g(\omega))$（参见 Lebeau [Math. Phys. Stud. 19 (1996) 73-109]）。 这里，$\mu(\omega)$表示阻尼波动方程算子的谱吸收值，$g(\omega)$是一个称为$\omega$的几何量，定义如下。 在$\Omega$中的一条光线是质量点在$\Omega$内自由运动并在边界上发生弹性反射所生成的轨迹。 这些反射遵循几何光学定律。 于是，几何量$g(\omega)$被定义为平均轨迹长度的上极限（长时间渐近性）。 我们给出一个算法，当 $\omega$ 是有限个平方的并时，显式计算 $g(\omega)$。 显示英文摘要

摘要： 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.