Eigenmodes of the damped wave equation and small hyperbolic subsets

TL;DR

This paper investigates the behavior of high-frequency solutions to the damped wave equation on compact manifolds, showing they cannot concentrate near small hyperbolic sets, and establishes spectral gaps under certain conditions.

Contribution

It proves that high-frequency damped solutions avoid small hyperbolic neighborhoods and demonstrates the existence of spectral gaps under pressure conditions.

Findings

High-frequency solutions cannot concentrate near small hyperbolic sets.

Existence of an inverse logarithmic spectral gap under pressure conditions.

Spectral properties are linked to the geometry of undamped trajectories.

Abstract

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of $\beta$-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of $\beta$-damped trajectories of the geodesic flow. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.