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 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 $n\in \mathbb{N}$.

38 pages

References in corpus (5)

• Unique determination of a time-dependent potential for wave equations from partial data
• Optimality of increasing stability for an inverse boundary value problem
• Examples of exponential instability for elliptic inverse problems
• Simultaneously recovering potentials and embedded obstacles for anisotropic fractional Schrödinger operators
• Global uniqueness for the semilinear fractional Schrödinger equation

Cited by in corpus (4)

• Inverse problem for a nonlocal diffuse optical tomography equation
• Low regularity theory for the inverse fractional conductivity problem
• The fractional $p$-biharmonic systems: optimal Poincaré constants, unique continuation and inverse problems
• An inverse problem for semilinear equations involving the fractional Laplacian