Search

We study an inverse problem for the fractional wave equation with a potential by the measurement taking on arbitrary subsets of the exterior in the space-time domain. We are interested in the issues of uniqueness and stability estimate in the determination of the potential by the exterior Dirichlet-to-Neumann map. The main tools are the qualitative and quantitative unique continuation properties for the fractional Laplacian. For the stability, we also prove that the log type stability estimate is optimal. The log type estimate shows the striking difference between the inverse problems for the fractional and classical wave equations in the stability issue. The results hold for any spatial dimension n ∈ ℕ. [ABSTRACT FROM AUTHOR]

In this article we study an inverse problem for the space-time fractional parabolic operator (∂t-∆)s + Q with 0 < s < 1 in any space dimension. We uniquely determine the unknown bounded potential Q from infinitely many exterior Dirichlet-to-Neumann type measurements. This relies on Runge approximation and the dual global weak unique continuation properties of the equation under consideration. In discussing weak unique continuation of our operator, a main feature of our argument relies on a new Carleman estimate for the associated degenerate parabolic Caffarelli- Silvestre extension. Furthermore, we also discuss constructive single measurement results based on the approximation and unique continuation properties of the equation. [ABSTRACT FROM AUTHOR]

In this work, we use monotonicity-based methods for the fractional Schrödinger equation with general potentials q Є L∞ (Ω) in a Lipschitz bounded open set Ω\subset Rn in any dimension n Є N. We demonstrate that if-and-only-if monotonicity relations between potentials and the Dirichlet-to-Neumann map hold up to a finite dimensional subspace. Based on these if-and-only-if monotonicity relations, we derive a constructive global uniqueness result for the fractional Calder\'on problem and its linearized version. We also derive a reconstruction method for unknown obstacles in a given domain that only requires the background solution of the fractional Schrödinger equation, and we prove uniqueness and Lipschitz stability from finitely many measurements for potentials lying in an a priori known bounded set in a finite dimensional subset of L∞ (Ω). [ABSTRACT FROM AUTHOR]

We consider an inverse problem for the fractional Schrödinger equation by using monotonicity formulas. We provide if-and-only-if monotonicity relations between positive bounded potentials and their associated nonlocal Dirichlet-to-Neumann maps. Based on the monotonicity relation, we can prove uniqueness for the nonlocal Calderón problem in a constructive manner. Second, we offer a reconstruction method for unknown obstacles in a given domain. Our method is independent of the dimension and only requires the background solution of the fractional Schrödinger equation. [ABSTRACT FROM AUTHOR]

We consider field localizing and concentration of electromagnetic waves governed by the time-harmonic anisotropic Maxwell system in a bounded domain. It is shown that there always exist certain boundary inputs which can generate electromagnetic fields with the energy localized/concentrated in a given subdomain while nearly vanishing in another given subdomain. The theoretical results may have potential applications in telecommunication, inductive charging, and medical therapy. We also derive a related Runge approximation result for the time-harmonic anisotropic Maxwell system with partial boundary data. [ABSTRACT FROM AUTHOR]

We develop an enclosure-type reconstruction scheme to identify penetrable and impenetrable obstacles in an electromagnetic field with anisotropic medium in R³. The main difficulty in treating this problem lies in the fact that there are so far no complex geometrical optics solutions available for Maxwell's equation with anisotropic medium in R³. Instead, we derive and use another type of special solutions called oscillating-decaying solutions. To justify this scheme, we use Meyers' Lp estimate, for the Maxwell system, to compare the integrals coming from oscillating-decaying solutions and those from the reflected solutions. [ABSTRACT FROM AUTHOR]