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. doi: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 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.