Global uniqueness for a 3D/2D inverse conductivity problem via the modified method of Carleman estimates

Abstract

Let Ω ⊂ be a bounded domain and {( x 1 , x 2 , x 3 ) | ( x 1 , x 3 ) ∈ Ω, x 2 ∈ (−∞, ∞)} be a cylinder. Assume that the stationary electric field u induced by a pointwise current modelled by the function f ( x 2 ) S ( x 1 , x 3 ) is governed by an elliptic differential equation ∇ · (σ∇ u ) = − f ( x 2 ) S ( x 1 , x 3 ). Also, assume that a pair of the Dirichlet and Neumann conditions is given on an arbitrary part of the lateral surface of the cylinder. The problem of determining the 2D electric conductivity σ( x 1 , x 3 ) in this cylinder from the boundary incomplete data is considered. Using the direct Fourier and inverse Laplace transforms, the original inverse problem is reduced to an auxiliary inverse problem for a hyperbolic equation . The main difficulty is due to the presence of derivatives of the function σ ( x 1 , x 3 ). To overcome this difficulty, the method of Carleman estimates is significantly modified. This allows for establishing the Lipschitz stability for the auxiliary hyperbolic inverse problem. In turn, the stability result implies the global uniqueness theorem for the original inverse conductivity problem. The proposed formulation and uniqueness theorem extend both the formulation of the 1D inverse problem of electrical prospecting and uniqueness result established by Tikhonov in 1949.