On the stability of an inverse problem for the wave equation

Consider the inverse problem of determining the potential q from the Neumann to Dirichlet map Lambda(q) of the wave equation u(tt) - Delta u + qu = 0 in Omega x (0, T) with u(x, 0) = u(t) (x, 0) = 0. In this paper, a nearly Lipschitz-type stability estimate is established for the inverse problem: for any small epsilon > 0, there exists beta(0) > 0 such that parallel to q(1) - q(2)parallel to (infinity)(L)((Omega)) <= C parallel to Lambda(q1) - Lambda(q2) parallel to(1-epsilon)(*) when parallel to q(1) - q(2)parallel to(H beta (Rn)) <= M for some beta > beta(0). Here, parallel to.parallel to(*) represents the operator norm.