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INVERSE PROBLEMS FOR THIRD-ORDER NONLINEAR PERTURBATIONS OF BIHARMONIC OPERATORS

by S. Bhattacharya and K. Krupchyk and S. Sahoo and G. Uhlmann

(Report number 2023-44)

Abstract
We study inverse boundary problems for third-order nonlinear tensorial perturbations of biharmonic operators on a bounded domain in \(\mathbb{R}^n\), where \(n\ge 3\). 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.

Keywords: Perturbed biharmonic operator; third order anisotropic perturbation; Tensor tomography; Momentum ray transform.

@Techreport{ KSU23_1081,
  author = {S.  Bhattacharya and K. Krupchyk and S. Sahoo and G. Uhlmann},
  title = {INVERSE PROBLEMS FOR THIRD-ORDER NONLINEAR PERTURBATIONS OF BIHARMONIC OPERATORS},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2023-44},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2023/2023-44.pdf },
  year = {2023}
}

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