Uniqueness principle for fractional (non)-coercive anisotropic polyharmonic operators and applications to inverse problems

In this work, we are concerned with inverse problems involving anisotropic poly-fractional operators, where the poly-fractional operator is of the form$ P( (-\Delta_g)^s)u : = \sum\limits_{i = 1}^M \alpha_i(-\Delta_{g_i})^{s_i}u $for $ s = (s_1,\dots,s_M) $, $ 0<s_1<\cdots<s_M<\infty $, $ s_M\in\mathbb{R}_+\backslash\mathbb{Z} $, $ g = (g_1,\dots,g_M) $, and sufficiently regular coefficients $ \alpha_i(x) $. There are three major contributions in this work that are new to the literature. First, we propose equations involving such anisotropic poly-fractional operators $ P $, which have not been previously considered in the general setting. Such equations arise naturally from the superposition of multiple stochastic processes with different scales, including classical random walks and Lévy flights. Secondly, we give novel results for the unique continuation properties for $ u $ when it is fractional polyharmonic, in the sense that $ u $ satisfies $ \tilde{P}((-\Delta_{\tilde{g}})^{\tilde{s}})u = 0 $ in a bounded Lipschitz domain $ \Omega $ for some $ \tilde{P} $. Our unique continuation property holds for relatively general $ \tilde{P} $, which is also anisotropic and in addition may not necessarily be coercive. With these results in hand, we consider the inverse problems for $ P $, and proved the uniqueness in recovering the potential, the source function in the semilinear case, and the coefficients associated to the non-isotropy of the fractional operator.