Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

By Giovanni Covi, Keijo Monkkonen, Jesse Railo. Inverse Problems and Imaging, 2020

58 citations recorded by the source corpus.

We prove a unique continuation property for the fractional Laplacian $(-\Delta)^s$ when $s \in (-n/2,\infty)\setminus \mathbb{Z}$. In addition, we study Poincar\'e-type inequalities for the operator $(-\Delta)^s$ when $s\geq 0$. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schr\"odinger equation. We also study the higher order fractional Schr\"odinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the $d$-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/40d24be833b0f81bf56d22e92b6f41bcdc11e087

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Citations in the corpus

The higher order fractional Calderón problem for linear local operators: Uniqueness (2020, background): “In the recent work [10], the first three authors of this article studied higher order versions (s ∈ R+ \Z) of the fractional Calderón problem and proved uniqueness for the Calderón problem for the fractional magnetic Schrödinger equation (up to a gauge).”