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Strong unique continuation for the higher order fractional Laplacian

Max-Planck-Institute for Mathematics in the Sciences, Inselstr. 22, 04103 Leipzig, Germany

† This contribution is part of the Special Issue: Inverse problems in imaging and engineering science
   Guest Editors: Lauri Oksanen; Mikko Salo
   Link: https://www.aimspress.com/newsinfo/1270.html

Special Issues: Inverse problems in imaging and engineering science

In this article we study the strong unique continuation property for solutions of higher order (variable coefficient) fractional Schrödinger operators. We deduce the strong unique continuation property in the presence of subcritical and critical Hardy type potentials. In the same setting, we address the unique continuation property from measurable sets of positive Lebesgue measure. As applications we prove the antilocality of the higher order fractional Laplacian and Runge type approximation theorems which have recently been exploited in the context of nonlocal Calderón type problems. As our main tools, we rely on the characterisation of the higher order fractional Laplacian through a generalised Caffarelli-Silvestre type extension problem and on adapted, iterated Carleman estimates.
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Keywords unique continuation; fractional Schrödinger equation; higher order nonlocal operators; Carleman estimates

Citation: María Ángeles García-Ferrero, Angkana Rüland. Strong unique continuation for the higher order fractional Laplacian. Mathematics in Engineering, 2019, 1(4): 715-774. doi: 10.3934/mine.2019.4.715

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