Low Regularity Ray Tracing for Wave Equations with Gaussian beams

TL;DR

This paper establishes observability estimates for wave equations with variable coefficients in higher dimensions using Gaussian beams, extending previous results and simplifying the construction of solutions.

Contribution

It generalizes observability estimates to higher dimensions and time-dependent coefficients, improving Gaussian beam construction for $C^{1,1}$ wave equations.

Findings

Proves observability estimates for wave equations with variable coefficients

Extends results to higher dimensions and time-dependent coefficients

Simplifies Gaussian beam construction for $C^{1,1}$ wave equations

Abstract

We prove observability estimates for oscillatory Cauchy data modulo a small kernel for $n$-dimensional wave equations with space and time dependent $C^2$ and $C^{1,1}$ coefficients using Gaussian beams. We assume the domains and observability regions are in $\mathbb{R}^n$, and the GCC applies. This work generalizes previous observability estimates to higher dimensions and time dependent coefficients. The construction for the Gaussian beamlets solving $C^{1,1}$ wave equations represents an improvement and simplification over Waters (2011).