The fractional Calderón problem : Low regularity and stability
Rüland, A., & Salo, M. (2020). The fractional Calderón problem : Low regularity and stability. Nonlinear Analysis: Theory, Methods and Applications, 193, 111529. https://doi.org/10.1016/j.na.2019.05.010
Published inNonlinear Analysis: Theory, Methods and Applications
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© 2019 Elsevier Ltd
The Calderón problem for the fractional Schrödinger equation was introduced in the work Ghosh et al. (to appear)which gave a global uniqueness result also in the partial data case. This article improves this result in two ways. First, we prove a quantitative uniqueness result showing that this inverse problem enjoys logarithmic stability under suitable a priori bounds. Second, we show that the results are valid for potentials in scale-invariant Lp or negative order Sobolev spaces. A key point is a quantitative approximation property for solutions of fractional equations, obtained by combining a careful propagation of smallness analysis for the Caffarelli–Silvestre extension and a duality argument.
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Related funder(s)European Commission; Academy of Finland
Funding program(s)FP7 (EU's 7th Framework Programme); Centre of Excellence, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingA.R. gratefully acknowledges a Junior Research Fellowship at Christ Church. M.S. was supported by the Academy of Finland (Finnish Centre of Excellence in Inverse Problems Research, grant number 284715 ) and by the European Research Council under FP7/2007–2013 ( ERC StG 307023 ) and Horizon 2020 ( ERC CoG 770924 ).
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Ghosh, Tuhin; Salo, Mikko; Uhlmann, Gunther (Mathematical Sciences Publishers, 2020)We show global uniqueness in an inverse problem for the fractional Schrödinger equation: an unknown potential in a bounded domain is uniquely determined by exterior measurements of solutions. We also show global uniqueness ...
Covi, Giovanni (Pergamon Press, 2020)This paper shows global uniqueness in two inverse problems for a fractional conductivity equation: an unknown conductivity in a bounded domain is uniquely determined by measurements of solutions taken in arbitrary open, ...
Rüland, Angkana; Salo, Mikko (American Institute of Mathematical Sciences, 2020)In this article we analyse quantitative approximation properties of a certain class of nonlocal equations: Viewing the fractional heat equation as a model problem, which involves both local and nonlocal pseudodifferential ...
Rüland, Angkana; Salo, Mikko (Institute of Physics, 2018)In this paper we prove the exponential instability of the fractional Calderón problem and thus prove the optimality of the logarithmic stability estimate from Rüland and Salo (2017 arXiv:1708.06294). In order to infer this ...
Lassas, Matti; Liimatainen, Tony; Lin, Yi-Hsuan; Salo, Mikko (Elsevier, 2021)We introduce a method for solving Calderón type inverse problems for semilinear equations with power type nonlinearities. The method is based on higher order linearizations, and it allows one to solve inverse problems for ...