Stabilization of the wave equation on larger-dimension tori with rough dampings
Abstract
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-G\'erard 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 . In some particular cases, we show that this sufficient condition can be weakened.
Cite
@article{arxiv.2303.03733,
title = {Stabilization of the wave equation on larger-dimension tori with rough dampings},
author = {Marc Rouveyrol},
journal= {arXiv preprint arXiv:2303.03733},
year = {2024}
}
Comments
32 pages, 7 figures. Accepted in Pure and Applied Analysis. Text overlap with arXiv:1801.00983 --- Updates in v2 : extended bibliography. Updates in v3 : explicit statement of the main result of Section 5 added (Theorem 5 of the introduction). Sections 1 and 5 modified accordingly. Corollary 2.2 made more precise. Other minor corrections. --- Updates in v4 : revisions based on referee remarks