A H\"older Stability Estimate for a 3D Coefficient Inverse Problem for a Hyperbolic Equation With a Plane Wave

TL;DR

This paper establishes a novel H"older stability estimate for a 3D coefficient inverse problem involving a hyperbolic equation with plane wave data, using Carleman estimates and finite difference techniques.

Contribution

It introduces the first H"older stability estimate for this inverse problem and provides new bounds on the solution's expansion near the characteristic wedge.

Findings

H"older stability estimates were successfully derived.

A new lower bound for the amplitude of the solution's first term was established.

Finite difference methods were applied to analyze the associated operator.

Abstract

A 3D coefficient inverse problem for a hyperbolic equation with non-overdetermined data is considered. The forward problem is the Cauchy problems with the initial condition the delta function concentrated at a single plane (i.e. the plane wave). A certain associated operator is written in finite differences with respect to two out of three spatial variables, i.e. "partial finite differences". The grid step size is bounded from the below by a fixed number. A Carleman estimate is applied to obtain, for the first time, a H\"older stability estimates for this problem. Another new result is an estimate from the below of the amplitude of the first term of the expansion of the solution of the forward problem near the characteristic wedge.