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

Abstract

<p style='text-indent:20px;'>We prove a unique continuation property for the fractional Laplacian <inline-formula><tex-math id="M1111">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M2">\begin{document}$ s \in (-n/2, \infty)\setminus \mathbb{Z} $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M3">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>. In addition, we study Poincaré-type inequalities for the operator <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^s $\end{document}</tex-math></inline-formula> when <inline-formula><tex-math id="M5">\begin{document}$ s\geq 0 $\end{document}</tex-math></inline-formula>. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the <inline-formula><tex-math id="M6">\begin{document}$ d $\end{document}</tex-math></inline-formula>-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.