Inverse Problems For Third-Order Nonlinear Perturbations Of Biharmonic Operators

TL;DR

This paper proves that the boundary measurements uniquely determine certain nonlinear third-order tensor perturbations of biharmonic operators, advancing inverse boundary problem theory in higher-order nonlinear PDEs.

Contribution

It establishes unique identifiability of nonlinear third-order tensor perturbations from boundary data, using generalized momentum ray transforms, for biharmonic operators.

Findings

Unique determination of nonlinear tensor perturbations from boundary data

Application of generalized momentum ray transforms to inverse problems

Addresses a previously open problem for nonlinear perturbations

Abstract

We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in $\mathbb{R}^n$, where $n\geq 3$. By imposing appropriate assumptions on the nonlinearity, we demonstrate that the Dirichlet-to-Neumann map, known on the boundary of the domain, uniquely determines the genuinely nonlinear tensorial third-order perturbations of the biharmonic operator. The proof relies on the inversion of certain generalized momentum ray transforms on symmetric tensor fields. Notably, the corresponding inverse boundary problem for linear tensorial third-order perturbations of the biharmonic operator remains an open question.