Exponential stabilization of waves for the Zaremba boundary condition

TL;DR

This paper establishes exponential stabilization of the wave equation with Zaremba boundary conditions under certain geometric assumptions, using semiclassical measures to analyze resolvent estimates and propagation near boundary points.

Contribution

It introduces novel techniques to handle measure support near boundary jumps and propagation near Neumann boundary points, advancing the understanding of wave stabilization with mixed boundary conditions.

Findings

Proves resolvent estimates on the imaginary axis

Demonstrates exponential stabilization under geometric conditions

Develops new semiclassical measure techniques for boundary analysis

Abstract

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.