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

摘要

We consider a restricted Dirichlet-to-Neumann map Lambda(T)(S, R) associated with the operator partial derivative(2)(t) - Delta(g) + A + q where Delta(g) 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 Lambda(T)(S, R) corresponds to the case where the Dirichlet traces are supported on (0, T) x S and the Neumann traces are restricted on (0, T) x R. Here S and R are open sets, which may be disjoint, on the boundary of M. We show that Lambda(T)(S, R) 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. Crown Copyright (C) 2019 Published by Elsevier Inc. All rights reserved.