Unique recovery of lower order coefficients for hyperbolic equations from data on disjoint sets

Abstract

We consider a restricted Dirichlet-to-Neumann map ΛS,RT associated with the operator ∂t2−Δg+A+q where Δg is the Laplace-Beltrami operator of a Riemannian manifold (M,g), and A and q are a vector field and a function on M. The restriction ΛS,RT corresponds to the case where the Dirichlet traces are supported on (0,T)×S and the Neumann traces are restricted on (0,T)×R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that ΛS,RT determines uniquely, up the natural gauge invariance, the lower order terms A and q in a neighborhood of the set R assuming that R is strictly convex and that the wave equation is exactly controllable from S in time T/2. We give also a global result under a convex foliation condition. The main novelty is the recovery of A and q when the sets R and S are disjoint. We allow A and q to be non-self-adjoint, and in particular, the corresponding physical system may have dissipation of energy.