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Stabilization of the wave equation on larger-dimension tori with rough dampings

Marc Rouveyrol

Vol. 6 (2024), No. 2, 487–520

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 and 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 3. In some particular cases, we show that this sufficient condition can be weakened.

wave equation, stabilization, geometric control, second microlocalization, third microlocalization
Mathematical Subject Classification
Primary: 93C20
Secondary: 35A27
Received: 29 March 2023
Revised: 16 January 2024
Accepted: 18 March 2024
Published: 16 May 2024
Marc Rouveyrol
Laboratoire de Mathématiques d’Orsay
Université Paris-Saclay