Momentum Ray Transforms and a Partial Data Inverse Problem for a Polyharmonic Operator

Abstract

In this article, we study an inverse problem with local data for a linear polyharmonic operator with several lower order tensorial perturbations. We consider our domain to have an inaccessible portion of the boundary where neither the input can be prescribed nor the output can be measured. We prove the unique determination of all the tensorial coefficients of the operator from the knowledge of the Dirichlet and Neumann map on the accessible part of the boundary, under suitable geometric assumptions on the domain.Assumption 1.1. We assume that 0 is not an eigenvalue of the operator L(x, D) : D(L(x, D)) → L 2 (Ω).

Local data inverse problem for the polyharmonic operator with anisotropic perturbations

“…In [Yan14], authors considered domains with inaccessible part of the boundary probed with a biharmonic operator with just first and a zeroth order perturbation. On the other hand, in [BKS23], the authors have considered polyharmonic operators with several lower order tensorial perturbations while the entire boundary is accessible. In this article, we gen-eralise the existing results by considering several tensorial perturbations as in (1.1) and an inaccessible part of the boundary Γ 0 = ∂Ω \ Γ, assuming that Γ 0 is part of a hyperplane.…”

“…Upon further analysis of these solutions, we reduce our problem to an uniqueness question for a series of momentum ray transforms (MRT) of the unknown tensorial perturbations. Recently in [BKS23,SS23] MRT has been proved to be useful for recovering higher order tensors from integral equations. We use uniqueness results of MRT from [BKS23,SS23] to finally obtain uniqueness of the tensorial perturbations and thus complete the proof of theorem 2.…”

“…Recently in [BKS23,SS23] MRT has been proved to be useful for recovering higher order tensors from integral equations. We use uniqueness results of MRT from [BKS23,SS23] to finally obtain uniqueness of the tensorial perturbations and thus complete the proof of theorem 2.…”

“…For a detailed description of semiclassical analysis and Poisson bracket present here, see [Mar02,Zwo12]. We recall the following Carleman estimate from [BKS23].…”

“…The study of coefficient recovery problems for biharmonic and polyharmonic operators was initiated by Krupchyk, Lassas, and Uhlmann in [KLU12b,KLU14]. It was soon followed by [Yan14,GK16,BG19a,BKS23,SS23], where coefficient recovery for biharmonic and polyharmonic operators in a bounded domain with dimensions n ⩾ 3 have been discussed. The higher order elliptic operators, as represented by (1.1), have received significant attention in various areas, such as physics and geometry, due to their connection with the theory of elasticity equation on thin domains given by the Kirchhoff plate equation (perturbed biharmonic operator) and the study of the Paneitz-Branson equation in conformal geometry.…”