Inverse problems for first-order hyperbolic equations with time-dependent coefficients

TL;DR

This paper establishes global Lipschitz stability for inverse problems involving first-order hyperbolic equations with space and time-dependent coefficients, using a novel Carleman estimate based on integral curves.

Contribution

It introduces a new approach to construct weight functions for Carleman estimates using integral curves of a vector field, enhancing stability analysis for hyperbolic inverse problems.

Findings

Proves Lipschitz stability for inverse source and coefficient problems

Develops a novel Carleman estimate using integral curves

Handles coefficients depending on both space and time

Abstract

We prove global Lipschitz stability for inverse source and coefficient problems for first-order linear hyperbolic equations, the coefficients of which depend on both space and time. We use a global Carleman estimate, and a crucial point, introduced in this paper, is the choice of the length of integral curves of a vector field generated by the principal part of the hyperbolic operator to construct a weight function for the Carleman estimate. These integral curves correspond to the characteristic curves in some cases.