Determination of a coefficient in the wave equation with a single measurement

We consider an inverse problem of finding the coefficient of the second-order derivatives in a second-order hyperbolic equation with variable coefficients. Under a weak regularity assumption and a geometrical condition on the metric, we prove the uniqueness in a multidimensional hyperbolic inverse problem with a single measurement of Neumann data on a suitable sub-boundary. Moreover we show that our uniqueness yields the Lipschitz stability estimate in L(2) space for solution to the inverse problem. The key is a Carleman estimate for a hyperbolic operator with variable coefficients.