Observability estimates for the wave equation with rough coefficients

The goal of this note is to prove observability estimates for the wave equation with a density which is only continuous in the domain, and satisfies some multiplier-type condition only in the sense of distributions. Our main argument is that one can construct suitable approximations of such density by a sequence of smooth densities whose corresponding wave equations are uniformly observable. The end of the argument then consists in a rather standard passage to the limit. (C) 2017 Academie des sciences. Published by Elsevier Masson SAS.