On the Sensitivity of Solutions of Hyperbolic Equations to the Coefficients

The goal of this work is to determine appropriate domain and range of the map from the coe cients to the solutions of the wave equation for which its linearization or formal derivative is bounded and the properties of the coe cients on which the bound depends. Such information is indispensable in the study of the inverse (coe cient identi cation) problem via smooth optimization methods. The main result of this paper is an explicit microlocal Sobolev estimate for the linearized forward map. In view of results of Rakesh [19] for the smooth coe cient case, the order of our regularity result is optimal. Our proof is based on the method of nonsmooth microlocal analysis, in particular various results on propagation of singularities, the method of progressing wave expansions, microlocal study of solutions of the transport equations, study of conormal properties of the fundamental solution, and a duality technique.