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 $n\in \mathbb{N}$.

arXiv comment38 pages

References in corpus (16)

• An extension problem related to the fractional Laplacian
• Ten equivalent definitions of the fractional Laplace operator
• Some observations on the Green function for the ball in the fractional Laplace framework
• Uniqueness and reconstruction for the fractional Calderón problem with a single measurement
• The Calderón problem for the fractional Schrödinger equation
• The fractional Calderón problem: low regularity and stability
• Unique Continuation for Fractional Schrödinger Equations with Rough Potential
• A new approach to hyperbolic inverse problems
• Monotonicity-based inversion of the fractional Schrödinger equation I. Positive potentials
• Exponential instability in the fractional Calderón problem
• Monotonicity-based inversion of the fractional Schrödinger equation II. General potentials and stability
• 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