Conductivity imaging by the method of characteristics in the 1-Laplacian

We consider the problem of reconstruction of a sufficiently smooth planar conductivity from the knowledge of the magnitude vertical bar J vertical bar of one current density field inside the domain, and the corresponding voltage and current on a part of the boundary. Mathematically, we are led to the Cauchy problem for the the 1-Laplacian with partial data. Different from existing works, we show that the equipotential lines are characteristics in a first order quasilinear partial differential equation. The conductivity can be recovered in the region flown by the characteristics originating at parts of the boundary where the data are available. Numerical experiments show the feasibility of this alternative method.