带粗糙阻尼的高维环面上波动方程的稳定性
Stabilization of the wave equation on larger-dimension tori with rough dampings
AI总结：
本文将二维环面上带多边形特征函数型粗糙阻尼的波动方程稳定性结论推广到任意d≥3维环面，给出高维情形下稳定性成立的充分条件，并证明部分情形下该条件可弱化。
摘要：
本文研究带阻尼波动方程的一致稳定性问题。已知当流形为紧流形且阻尼连续时，几何控制条件是充要条件。对于阻尼为二维环面上若干多边形特征函数之和的情形，Burq-Gérard的结果表明，稳定性成立当且仅当每条测地线都与阻尼区域内部相交，或从两侧擦过阻尼多边形。我们将他们的结果自然推广，给出了任意维数$d\geq3$的环面上稳定性成立的一个充分条件。在某些特定情形下，我们证明该充分条件还可以被弱化。
英文摘要：
This paper deals with uniform stabilization of the damped wave equation. When the manifold is compact and the damping is continuous, the geometric control condition is known to be necessary and sufficient. In the case where the damping is a sum of characteristic functions of polygons on a two-dimensional torus, a result by Burq-Gérard states that stabilization occurs if and only if every geodesic intersects the interior of the damped region or razes damped polygons on both sides. We give a natural generalization of their result to a sufficient condition on tori of any dimension $d \geq 3$. In some particular cases, we show that this sufficient condition can be weakened.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
