The Calderón Problem for the Fractional Wave Equation: Uniqueness and Optimal Stability

Abstract

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 Dirichletto-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.

An inverse problem for semilinear equations involving the fractional Laplacian

“…In [KW23,Li22,Li23b], the authors considered inverse problems for generalized Kerr-type nonlinearity, which is different with ours. Then in [KLW22], the authors studied an inverse problem involving fractional wave equation, while in [LLL21], the authors solved an inverse problem for hyperbolic systems.…”

“…We use notations for fractional Sobolev spaces as in [KLW22]. To make the paper selfcontained, we give brief introductions to them.…”

Inverse problems for fractional equations with a minimal number of measurements

“…This is not unusual when one passes from finite unknowns to infinite unknowns in the study of inverse problems due to their ill-posed nature; see e.g. [2,41,25,20]. Instead, we prove a unique determination result by using uncountably many measurements in the case N = ∞.…”

“…It turns out that one can solve several challenging inverse problems that still remain unsolved in local cases, namely the fractional PDOs are replaced by the corresponding non-fractional counterparts, by taking advantage of the nonlocality of the fractional PDEs. We refer to [1,15,5,4,6,11,19,20,16,18,14,37,30,25,49,48] and the references cited therein for the existing developments in the literature.…”

The Global Uniqueness of a Dissipative Fractional Helmholtz Equation

“…Rland and Salo had provided the low regularity and stability for the fractional Calderón problem [7]. Covi had proved the uniqueness for the fractional Calderón problem with quasilocal perturbations [8]. Kow, Lin, and Wang had provided the uniqueness and optimal stability of the Calderón problem for the fractional wave equation [9].…”