Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

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

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 $Δ_g^2+q$ on a conformally transversally anisotropic Riemannian manifold of dimension $\ge 3$ 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 [51]. 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 $3$-dimen…

Uniqueness of a Potential from Boundary Data in Locally Conformally Transversally Anisotropic Geometries

Let $(Ω^3,g)$ be a compact smooth Riemannian manifold with smooth boundary and suppose that $U$ is a an open set in $Ω$ such that $g|_U$ is the Euclidean metric. Let $Γ= \overline{U} \cap \partial Ω$ be connected and suppose that $U$ is the convex hull of $Γ$. We will study the uniqueness of an unknown potential for the Schrödinger operator $ -\triangle_g + q $ from the associated Dirichlet to Neumann map, $Λ_q$. We will prove that if the potential $q$ is a priori explicitly known in $U^c$, then one can uniquely reconstruct $q$ over the convex hull of $Γ$ from $Λ_q$. We will also outline a rec…

The linearized Calderon problem in transversally anisotropic geometries

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic…

The Calderon problem in transversally anisotropic geometries

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Four…

Inverse problems for magnetic Schrödinger operators in transversally anisotropic geometries

We study inverse boundary problems for magnetic Schrödinger operators on a compact Riemannian manifold with boundary of dimension $\ge 3$. In the first part of the paper we are concerned with the case of admissible geometries, i.e. 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 Cauchy data on the boundary of the manifold for the magnetic Schrödinger operator with $L^\infty$ magnetic and electric potentials, determines the magnetic field and electric potential uniquely. In the …

双曲波逆散射问题的直接成像方法：带相位和无相位数据

A direct imaging method for inverse scattering problem of biharmonic wave with phased and phaseless data

本文提出基于反时间迁移的直接成像方法，用于恢复具有四种边界条件之一的扩展障碍物，利用单一测量数据类型进行障碍物形状和位置的识别。