Exponential instability in the fractional Calderón problem. (20th February 2018)

• Record Type: Journal Article
• Authors: Rüland, Angkana; Salo, Mikko
• Is Part Of: Inverse problems. Volume 34:Number 4(2018:Apr.)
• Publication Date: 2018-02-20
• Subjects: fractional Calderón problem -- exponential instability -- Runge approximation; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aaac5a
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: 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 result, we follow the strategy introduced by Mandache in (2001 Inverse Problems 17 1435) for the standard Calderón problem. Here we exploit a close relation between the fractional Calderón problem and the classical Poisson operator. Moreover, using the construction of a suitable orthonormal basis, we also prove (almost) optimality of the Runge approximation result for the fractional Laplacian, which was derived in Rüland and Salo (2017 arXiv:1708.06294). Finally, in one dimension, we show a close relation between the fractional Calderón problem and the truncated Hilbert transform.