Determining the first order perturbation of a polyharmonic operator on admissible manifolds

Abstract

We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator Lg,X,q, with X being a W1,∞ vector field and q being an L∞ function on compact Riemannian manifolds with boundary which are conformally embedded in a product of the Euclidean line and a simple manifold. We show that the knowledge of the Dirichlet-to-Neumann map determines X and q uniquely. The method is based on the construction of complex geometrical optics solutions using the Carleman estimate for the Laplace–Beltrami operator due to Dos Santos Ferreira, Kenig, Salo and Uhlmann. Notice that the corresponding uniqueness result does not hold for the first order perturbation of the Laplace–Beltrami operator.