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Arefinement of the Bukhgeim-Klibanov method

Suliang Si

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.

Arefinement of the Bukhgeim-Klibanov method

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.

Paper Structure

This paper contains 3 sections, 2 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Carleman estimates
  3. Proof of Theorem \ref{['Thm']}

Key Result

Theorem 1.1

Let $T > T_0$ and $(u_0, u_1) \in H^2(\Omega)\times H^1(\Omega)$ , $h\in H^2(0,T;H^2(\Omega))$. We assume that there exist constant $M_0,\ m_0 > 0$ such that Then there exists a constant $C > 0$ such that Here the constant $C$ is dependent on $\Omega$, $T$, $m_0$, $M_0$.

Theorems & Definitions (4)

  • Theorem 1.1
  • Lemma 2.1
  • Remark 2.2
  • proof