Stability estimate for an inverse problem for the wave equation in a magnetic field

Abstract

In this article, we prove stability estimate of the inverse problem of determining the magnetic field entering the magnetic wave equation in a bounded smooth domain in ℝ ^d from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the magnetic wave equation. We prove in dimension d ≥ 2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic wave equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.

Keywords:

• stability estimate
• hyperbolic inverse problem
• magnetic field
• Dirichlet-to-Neumann-map

Acknowledgement

The authors thank the anonymous referee for very useful comments.