Concentration Of Laplace Eigenfunctions And Stabilization Of Weakly Damped Wave Equation

TL;DR

This paper establishes universal bounds on the concentration of Laplace eigenfunctions near submanifolds and derives new decay rate results for weakly damped wave equations on compact manifolds.

Contribution

It introduces new bounds on eigenfunction concentration and links these to improved decay estimates for weakly damped wave equations.

Findings

Universal bounds on eigenfunction concentration near submanifolds.

New decay rate estimates for weakly damped wave equations.

Enhanced understanding of eigenfunction behavior in relation to damping.

Abstract

- In this article, we prove some universal bounds on the speed of concentration on small (frequency-dependent) neighborhoods of submanifolds of L 2-norms of quasi modes for Laplace operators on compact manifolds. We deduce new results on the rate of decay of weakly damped wave equations. R{\'e}sum{\'e}.