Inverse Boundary Problems for Biharmonic Operators in Transversally Anisotropic Geometries

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

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We prove that a continuous potential q can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the perturbed biharmonic operator ∆ 2 g + q on a conformally transversally anisotropic Riemannian manifold of dimension ≥ 3 with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [56]. In particular, our result is applicable and new in the case of smooth bounded domains in the 3-dimensional Euclidean space as well as in the case of 3-dimensional admissible manifolds.

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“…Our main result is as follows, and it gives a constructive counterpart to the uniqueness result of [56].…”

“…Going beyond the Euclidean setting, the global uniqueness in the inverse boundary problem for zero and first-order perturbations of the biharmonic operator was only obtained in the case when the manifold (M, g) is admissible in [5], see Definition 1.2 below, and in the more general case when (M, g) is CTA (conformally transversally anisotropic, see Definitions 1.1) with the injective geodesic X-ray transform on the transversal manifold (M 0 , g 0 ) in [56]. The works [5] and [56] are extensions of the fundamental works [16] and [17] which initiated this study in the case of perturbations of the Laplacian. We refer to the works [39], [21], [20], [38], for inverse boundary problems for nonlinear Schrödinger equations on CTA manifolds, and we remark that there are no assumptions on the transversal manifold in these works.…”

“…The proofs of the global uniqueness results in the works [16], [17] [5], [56] rely on construction of complex geometric optics solutions based on the techniques of Carleman estimates with limiting Carleman weights. Thanks to the work [16], we know that the property of being a CTA manifold guarantees the existence of limiting Carleman weights.…”

Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

“…The most recent paper [7] establishes the uniqueness of a second order term from boundary data. Work of Yan [34] establishes uniqueness for first order perturbations of a biharmonic operator on transversally isotropic manifolds. She extends this work to give a constructive recovery procedure for continuous potentials [35].…”