Quantitative estimates of strong unique continuation for anisotropic wave equations

TL;DR

This paper provides quantitative estimates for solutions to anisotropic wave equations, establishing strong unique continuation properties that determine when solutions must vanish based on their behavior on certain segments or boundary portions.

Contribution

It introduces new quantitative estimates for anisotropic wave equations that strengthen the understanding of unique continuation properties beyond previous qualitative results.

Findings

Solutions vanish near flat segments if flatness is known.

Solutions vanish near boundary portions if flatness and boundary conditions are met.

Quantitative estimates extend previous qualitative unique continuation results.

Abstract

The main results of the present paper consist in some quantitative estimates for solutions to the wave equation $\partial^2_{t}u-\mbox{div}\left(A(x)\nabla_x u\right)=0$. Such estimates imply the following strong unique continuation properties: (a) if $u$ is a solution to the the wave equation and $u$ is flat on a segment $\{x_0\}\times J$ on the $t$ axis, then $u$ vanishes in a neighborhood of $\{x_0\}\times J$. (b) Let u be a solution of the above wave equation in $\Omega\times J$ that vanishes on a a portion $Z\times J$ where $Z$ is a portion of $\partial\Omega$ and $u$ is flat on a segment $\{x_0\}\times J$, $x_0\in Z$, then $u$ vanishes in a neighborhood of $\{x_0\}\times J$. The property (a) has been proved by G. Lebeau, Comm. Part. Diff. Equat. 24 (1999), 777-783.