Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

Brown, Russell M.; Gauthier, Luc

doi:10.3934/ipi.2022006

Cited by 4 publications

(3 citation statements)

References 34 publications

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“…If we extend our consideration to include first order perturbations, it was proved in [23] that the set of Cauchy data determines sufficiently smooth first order perturbation of the polyharmonic operator uniquely. There are extensive subsequent efforts to recover the first order perturbation of the polyharmonic operator with lower regularity, see for instance [4,5,10,24] and the references therein. We remark that relaxing the regularity of coefficients is crucial in inverse problems, since it enables the imaging of rough medium.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

Liu¹

2020

Inverse Problems &Amp; Imaging

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We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established in [37] and [7], respectively. We establish stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials, sharpening those of [7].

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

Liu¹

2020

Inverse Problems &Amp; Imaging

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“…If we extend our consideration to include first order perturbations, it was proved in [26] that the Dirichlet-to-Neumann map determines sufficiently smooth first order perturbation of the polyharmonic operator uniquely. There are extensive subsequent efforts to recover the first order perturbation of the polyharmonic operator with lower regularity, see for instance [5,6,11,27] and the references therein. We remark that relaxing the regularity of coefficients is crucial in inverse problems, since it enables the imaging of rough medium.…”

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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confidence: 99%

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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Inverse boundary value problems for polyharmonic operators with non-smooth coefficients

Cited by 4 publications

References 34 publications

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies

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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