Arefinement of the Bukhgeim-Klibanov method

TL;DR

This paper enhances the Bukhgeim-Klibanov method for inverse source problems in hyperbolic equations by introducing a new Carleman estimate that simplifies proofs and broadens applicability.

Contribution

The paper introduces a novel Carleman estimate that refines the Bukhgeim-Klibanov method, removing the need for time extension and simplifying stability proofs for hyperbolic inverse problems.

Findings

Developed a new Carleman estimate applicable to evolution equations.

Simplified the proof of conditional stability for inverse source problems.

Broadened the method's applicability to various hyperbolic equations.

Abstract

In this article, we improve the classical Bukhgeim-Klibanov method presented in [1],which can be used to prove the conditional stability of inverse source problem for a hyperbolic equation from the measurement on the subboundary. A major ingredient of our proof is a novel Carleman estimate. This inequality eliminates the need to extend the solution in time, therefore simplifies the existing proofs, which is widely applicable to various evolution equations.