Let A ∈ Sym(n × n) be an elliptic 2-tensor. Consider the anisotropic fractional Schrödinger operator L s A +q, where L s A := (−∇ · (A(x)∇)) s , s ∈ (0, 1) and q ∈ L ∞ . We are concerned with the simultaneous recovery of q and possibly embedded soft or hard obstacles inside q by the exterior Dirichlet-to-Neumann (DtN) map outside a bounded domain Ω associated with L s A + q. It is shown that a single measurement can uniquely determine the embedded obstacle, independent of the surrounding potential q. If multiple measurements are allowed, then the surrounding potential q can also be uniquely recovered. These are surprising findings since in the local case, namely s = 1, both the obstacle recovery by a single measurement and the simultaneous recovery of the surrounding potential by multiple measurements are longstanding problems and still remain open in the literature. Our argument for the nonlocal inverse problem is mainly based on the strong uniqueness property and Runge approximation property for anisotropic fractional Schrödinger operators.
We are concerned with an inverse problem associated with the fractional Helmholtz system that arises from the study of viscoacoustics in geophysics and thermoviscous modelling of lossy media. We are particularly interested in the case that both the medium parameter and the internal source of the wave equation are unknown. Moreover, we consider a general class of source functions which can be frequencydependent. We establish several general uniqueness results in simultaneously recovering both the medium parameter and the internal source by the corresponding exterior measurements. In sharp contrast, these unique determination results are unknown in the local case, which would be of significant importance in thermo-and photo-acoustic tomography.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.