Exponential stabilization of waves for the Zaremba boundary condition

Pierre Cornilleau , Luc Robbiano (1)

1 LMV - Laboratoire de Mathématiques de Versailles

In this article 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 technics. The main difficulties are to prove that support of measure is in characteristic set in a neighborhood of the jump in the boundary condition and to prove results of propagation in a neighborhood of a boundary point where Neumann boundary condition is imposed. In fact if a lot of results applied here are proved in previous articles, these two points are new.

Mots clés: pseudo-differential calculus; controllability; Stabilization of Waves; semiclassical measure; Zaremba problem; pseudo-differential calculus; boundary propagation; pseudo-differential calculus; pseudo-differential calculus

Domaines: Equations aux dérivées partielles [math.AP]

Pierre Cornilleau, Luc Robbiano. Exponential stabilization of waves for the Zaremba boundary condition. 2021. ⟨hal-03364288⟩