Unique determination of anisotropic perturbations of a polyharmonic operator from partial boundary data

Abstract

We study an inverse problem involving the unique recovery of several lower order anisotropic tensor perturbations of a polyharmonic operator in a bounded domain from the knowledge of the Dirichlet to Neumann map on a part of boundary. The uniqueness proof relies on the inversion of generalized momentum ray transforms (MRT) for symmetric tensor fields, which we introduce for the first time to study Calderón-type inverse problems. We construct suitable complex geometric optics (CGO) solutions for the polyharmonic operators that reduces the inverse problem to uniqueness results for a generalized MRT. The uniqueness result and the inversion formula we prove for generalized MRT could be of independent interest and we expect it to be applicable to other inverse problems for higher order operators involving tensor perturbations.

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

“…Next, we solve the transport equations (3.24) following [BKS21]. Observe that, up to a rotation, we can take µ (1) = e n and µ…”

“…Recovering the coefficients. In this section, we follow the induction method described in [BKS21] to recover the coefficients. We multiply (3.22) by suitable powers of h and take h → 0 to obtain momentum ray transforms of the coefficients.…”

“…Next we construct complex geometric optics solutions of the equation L(x, D)u = 0 based on Proposition 2.2. Following [GK16,BKS21], we propose a solution of L(x, D)u = 0 in the CGO form:…”

“…Moreover, the setup is even more complicated due to complex vectors in place of the real vectors ξ ∈ R n . In [BKS21], the authors define the momentum ray transform of…”

“…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,BG19,BKS21,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.…”