Strong unique continuation for the higher order fractional Laplacian

María Ángeles García-Ferrero; Angkana Rüland; María Ángeles García-Ferrero; Angkana Rüland

doi:10.3934/mine.2019.4.715

Mathematics in Engineering

2019, Volume 1, Issue 4: 715-774. doi: 10.3934/mine.2019.4.715

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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

Received: 26 February 2019 Accepted: 22 June 2019 Published: 20 August 2019

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.
- 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[J]. Mathematics in Engineering, 2019, 1(4): 715-774. doi: 10.3934/mine.2019.4.715

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Abstract

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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