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

Abstract

Abstract. We consider the inverse boundary value problem for the first order perturbation of the polyharmonic operator L g,X,q , with X being a W 1,∞ 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-toNeumann 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.

Inverse Problems for Magnetic Schrödinger Operators in Transversally Anisotropic Geometries

“…Finally, assuming that q (1) = q (2) , using parallel transport along loops in M , boundary reconstruction of the magnetic potential, and unique continuation arguments, as in [14] and [5], we show that the fluxes of the magnetic potentials A (1) and A (2) satisfy…”

“…It can be viewed as an analog of our previous result [23] in the Euclidean case, in the setting of admissible manifolds. (2) ,q (2) , then dA (1) = dA (2) and q (1) = q (2) .…”

“…Let (M, g) be a conformally transversally anisotropic manifold such that Assumption 1 holds for the transversal manifold. Let A (1) , A (2) ∈ C(M, T * M ) be complex valued 1-forms, and q (1) (2) ,q (2) , then dA (1) = dA (2) . Assuming furthermore that q (1) = q (2) , we have…”

“…Our next goal is to show that q (1) = q (2) in M . Since M is simply connected, by the Poincaré lemma for currents, see [33], we conclude that there is…”

The Attenuated Geodesic Ray Transform on Tensors: Generic Injectivity and Stability

“…In [11], unique determination of the conductivity from the boundary measurements was reduced to injectivity of I a . In a similar way the latter is related to inverse problems for other elliptic equations and systems [4,19,20,21] including nonlinear ones [5]. The transform I a arises in several problems as well.…”