Recovery of non-smooth coefficients appearing in anisotropic wave equations

Abstract

We study the problem of unique recovery of a non-smooth one-form $\mathcal{A}$ and a scalar function $q$ from the Dirichlet to Neumann map, ${\Lambda }_{\mathcal{A},q}$, of a hyperbolic equation on a Riemannian manifold $(M,g)$. We prove uniqueness of the one-form $\mathcal{A}$ up to the natural gauge, under weak regularity conditions on $\mathcal{A},q$ and under the assumption that $(M,g)$ is simple. Under an additional regularity assumption, we also derive uniqueness of the scalar function $q$. The proof is based on the geometric optic construction and inversion of the light ray transform extended as a Fourier Integral Operator to non-smooth parameters and functions.

References in corpus (2)

• The Inverse Problem for the Dirichlet-to-Neumann map on Lorentzian manifolds
• Unique determination of a time-dependent potential for wave equations from partial data

Cited by in corpus (3)

• The Light Ray transform in Stationary and Static Lorentzian geometries
• Global recovery of a time-dependent coefficient for the wave equation from a single measurement
• A uniqueness result for light ray transform on symmetric 2-tensor fields