Lipschitz stability of recovering the conductivity from internal current densities

Abstract

Abstract In recent years, coupled physics imaging techniques have been developed to produce clearer images than those produced by electrical impedance tomography. This paper focuses on the inverse problem arising in current density impedance imaging and magneto-acousto-electric tomography. We consider the electrostatic equation ∇ ⋅ ( σ ∇ w b ) = 0 in a bounded domain Ω ⊂ R 3 with either the Dirichlet or Neumann boundary condition b, where σ is a scalar conductivity function. The inverse problem is formulated as recovering σ from vector fields J b = σ ∇ w b with different boundary conditions b. We provide a local Lipschitz stability, stating that near some known σ 0 and under some regularity assumptions, we can find b 1 and b 2 by constructing complex geometrical optics (CGO) solutions such that ‖ ln σ ( 1 ) − ln σ ( 2 ) ‖ C m , α ( Ω ¯ ) ⩽ C ∑ j = 1 2 ‖ J b j ( 1 ) − J b j ( 2 ) ‖ C m , α ( Ω ¯ ) . Furthermore, we modify the CGO solutions using the reflection method to make b 1 and b 2 vanish on a portion of a plane, and prove a local Lipschitz stability with partial data.