Stability of the unique continuation for the wave operator via Tataru inequality and applications

TL;DR

This paper establishes a logarithmic stability estimate for the unique continuation of wave equations with variable coefficients, using Tataru's Carleman estimate, applicable in arbitrary domains and after conjugate points.

Contribution

It extends the stability results for wave equations by providing explicit quantitative estimates based on Tataru's inequality, applicable in complex geometric settings.

Findings

Logarithmic stability estimate for wave equations with variable coefficients

Explicit parameter dependence on coefficient regularity and geometry

Stable estimates valid after conjugate points in geodesic flow

Abstract

In this paper we study the stability of the unique continuation in the case of the wave equation with variable coefficients independent of time. We prove a logarithmic estimate in a arbitrary domain of ${\mathbb R}^{n+1}$, where all the parameters are calculated explicitly in terms of the $C^1$-norm of the coefficients and on the other geometric properties of the problem. We use the Carleman-type estimate proved by Tataru in 1995 and an iteration for locals stability. We apply the result to the case of a wave equation with data on a cylinder an we get a stable estimate for any positive time, also after the first conjugate point for the geodesics of the metric related to the variable coefficients.