An inverse source problem for Maxwell's equations in anisotropic media

Abstract

In this article, we consider nonstationary Maxwell's equations in an anisotropic medium in the (x ₁, x ₂, x ₃)-space, where equations of the divergences of electric and magnetic flux densities are also unknown. Then we discuss an inverse problem of determining the x ₃-independent components of the electric current density from observations on the plane x ₃ = 0 over a time interval. Our main aim is, study conditional stability in the inverse problem provided the permittivity and the permeability are independent of x ₃. The main tool is a new Carleman estimate.

Keywords:

• Maxwell's equations
• Anisotropic
• Carleman estimate
• Inverse source problems
• AMS 2000 Mathematics Subject Classifications: 35R30
• 35Q60
• 78A25
• 35L40

Acknowledgments

The authors thank Professor V. Isakov (Wichita State University, USA) for the valuable comments and the information for the example in section 1. The first author was supported by the Fujyu-kai (Tokyo, Japan). The second author was supported in part by grant 15340027 from the Japan Society for Promotion of Science and grant 15654015 from the Ministry of Education, Cultures, Sports and Technology.