Low frequency estimates and local energy decay for asymptotically euclidean Laplacians

TL;DR

This paper establishes low frequency resolvent estimates for asymptotically Euclidean Laplacians and explores their implications for local energy decay in various wave equations.

Contribution

It provides new resolvent estimates at low frequencies for long-range perturbations of the Euclidean Laplacian, with applications to energy decay in PDEs.

Findings

Proved low frequency resolvent estimates for asymptotically Euclidean Laplacians.

Applied these estimates to demonstrate local energy decay for Schrödinger, Wave, and Klein-Gordon equations.

Abstract

For Laplace-Beltrami operators associated to metrics which are long range perturbations of the flat one, we prove estimates for powers of the resolvent as the spectral parameter goes to zero. We also discuss applications to the local energy decay for the Schroedinger, Wave and Klein Gordon equations.