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Exponential stabilization of waves for the Zaremba boundary condition

Pierre Cornilleau and Luc Robbiano

Vol. 6 (2024), No. 1, 1–71
DOI: 10.2140/paa.2024.6.1

We prove, under some geometrical condition on geodesic flow, exponential stabilization of wave equation with Zaremba boundary condition. We prove an estimate on the resolvent of semigroup associated with wave equation on the imaginary axis and we deduce the stabilization result. To prove this estimate we apply semiclassical measure techniques. The main difficulties are proving that support of measure is in characteristic set in a neighborhood of the jump in the boundary condition and proving results of propagation in a neighborhood of a boundary point where Neumann boundary condition is imposed. In fact a lot of results applied here are proved in previous articles; these two points are new.

stabilization of waves, Zaremba problem, pseudodifferential calculus, controllability, semiclassical measure, boundary propagation
Mathematical Subject Classification
Primary: 35L20
Secondary: 35B35
Received: 6 November 2023
Accepted: 12 December 2023
Published: 22 February 2024
Pierre Cornilleau
Lycée Pothier
Luc Robbiano
Laboratoire de Mathématiques
Université de Versailles Saint-Quentin en Yvelines, UVSQ, CNRS