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

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 an arbitrary domain of Rn+1, where all the parameters are calculated explicitly in terms of the C1-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 of the local stability. We apply the result to the case of a wave equation with data on a cylinder and we get a stable estimate for any positive time, also after the first conjugate point associated with the geodesics of the metric of the variable coefficients.