Quantitative estimates of strong unique continuation for anisotropic wave equations

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.