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

Abstract

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].

Linearized inverse Schrödinger potential problem with partial data and its deep neural network inversion

“…When the inaccessible part of observation is a subset of a hyperplane, [23] showed the increasing stability for the inverse Schrödinger problem by extending the idea in [13] and calibrating the almost exponential solutions carefully. A stability estimate with explicit frequency dependence for perturbed biharmonic operator in the same partial boundary setting was obtained in [24]. We also mention the result in [21] for another frequencyexplicit increasing stability estimate in a different partial boundary observation setting.…”

“…The proof is similar to [24] and we briefly present it in Appendix B. Note that in [24] the author does not assume the potential function to be compactly supported in the domain Ω.…”

“…Compared to the classical methods of CGO solutions for complete boundary data [26], the Riemann-Lebesgue lemma plays an important role in [13] to eliminate the redundant Fourier modes produced by reflection, which should be sharpened to a quantitative version when deriving stability estimates. Various quantitative results have been obtained, depending on different a-priori assumptions on the potential functions, see [11] and [24]. In particular, functions in H s (R n ), 0 ≤ s < 1, are considered in [24] and estimates are derived using mollification.…”

“…Various quantitative results have been obtained, depending on different a-priori assumptions on the potential functions, see [11] and [24]. In particular, functions in H s (R n ), 0 ≤ s < 1, are considered in [24] and estimates are derived using mollification. That result remains valid for H 1 (R n ) functions with compact support, as shown below.…”

High‐frequency stability estimates for a partial data inverse problem

“…We would like to mention that the analysis of the behavior of the stability estimates from partial DN map as the frequency grows has been either known only in certain special type of geometries (see previous studies 31–33 ) or with impedance type boundary conditions and/or under the assumption of knowledge of the potential in a neighborhood of the boundary (see Garcia‐Ferrero et al 2 and Krupchyk and Uhlmann 34 ). In our work, we address this question for the partial data case considered by Bukhgeim and Uhlmann 15 and Heck and Wang 17 …”