An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

Bhattacharyya, Sombuddha; Ghosh, Tuhin

doi:10.1007/s00208-021-02276-6

Cited by 11 publications

(6 citation statements)

References 35 publications

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“…The inverse boundary problem that we are interested in is to determine the vector field X and the potential q from the knowledge of the set of the Cauchy data C X,q . This problem was studied extensively in the Euclidean setting, see [27], [28], [2], [3], [23] [24] [7], [6], [18], [19], [42]. Specifically, it was shown in [27] that the set of the Cauchy data C X,q determines the vector field X and the potential q uniquely.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

Yan¹

2020

Preprint

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We study inverse boundary problems for first order perturbations of the biharmonic operator on a conformally transversally anisotropic Riemannian manifold of dimension n ≥ 3. We show that a continuous first order perturbation can be determined uniquely from the knowledge of the set of the Cauchy data on the boundary of the manifold provided that the geodesic X-ray transform on the transversal manifold is injective.4 YAN complex vector fields, and let q (1) , q(1.4)we have q (1) = q (2) in M. Remark 1.3. Examples of non-simple manifolds M 0 satisfying Assumption 1 include in particular manifolds with strictly convex boundary which are foliated by strictly convex hypersurfaces [39], [41], and manifolds with a hyperbolic trapped set and no conjugate points [20], [21].Remark 1.4. To the best of our knowledge, Theorem 1.2 seems to be the first result where one recovers a vector field uniquely on general CTA manifolds.Remark 1.5. The assumption (1.4) is made for simplicity only and can be removed by performing the boundary determination as done in Appendix A for the vector fields X (1) and X (2) . This can be done by using the approach of [22] combined with its extensions in [32] and [16].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

Yan¹

2020

Preprint

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“…Since polyharmonic operators are of order 2m, it is natural to consider recovering second or higher order perturbations from boundary measurements, and some progress have been made in this direction. For instance, the authors of [8,9] obtained unique determination results for perturbations of up to second order appearing in the polyharmonic operator. Beyond the second order perturbations, it was proved in [10] that the Dirichlet-to-Neumann map, with the Neumann data measured on roughly half of the boundary, determines anisotropic perturbations of up to order m appearing in polyharmonic operators.…”

Section: Previous Literaturementioning

confidence: 99%

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

Liu

2024

Inverse Problems

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In this paper we study an inverse boundary value problem for the biharmonic operator with first order perturbation. Our geometric setting is that of a bounded simply connected domain in the Euclidean space of dimension three or higher. Assuming that the inaccessible portion of the boundary is flat, and we have knowledge of the Dirichlet-to-Neumann map on the complement, we prove logarithmic type stability estimates for both the first and the zeroth order perturbation of the biharmonic operator.

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“…While the work [28] considers the case of bounded potentials, certain classes of unbounded potentials are dealt with in the work [27], see also [35]. We refer to [33], [32] where the inverse boundary problem of determination of a first-order perturbation of the biharmonic operator was studied in the Euclidean case, see also [11], [1], [2], [4] for the case of non-smooth perturbations, and [9], [23] for the case of second order perturbations.…”

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

Yan¹

2023

IPI

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<p style='text-indent:20px;'>We prove that a continuous potential <inline-formula><tex-math id="M1">\begin{document}$ q $\end{document}</tex-math></inline-formula> can be constructively determined from the knowledge of the Dirichlet–to–Neumann map for the perturbed biharmonic operator <inline-formula><tex-math id="M2">\begin{document}$ \Delta_g^2+q $\end{document}</tex-math></inline-formula> on a conformally transversally anisotropic Riemannian manifold of dimension <inline-formula><tex-math id="M3">\begin{document}$ \ge 3 $\end{document}</tex-math></inline-formula> with boundary, assuming that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [<xref ref-type="bibr" rid="b56">56</xref>]. In particular, our result is applicable and new in the case of smooth bounded domains in the <inline-formula><tex-math id="M4">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional Euclidean space as well as in the case of <inline-formula><tex-math id="M5">\begin{document}$ 3 $\end{document}</tex-math></inline-formula>–dimensional admissible manifolds.</p>

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An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator

Cited by 11 publications

References 35 publications

Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

Inverse boundary problems for biharmonic operators in transversally anisotropic geometries

Stability estimates for an inverse boundary value problem for biharmonic operators with first order perturbation from partial data

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

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