Uniqueness of minimizers of weighted least gradient problems arising in hybrid inverse problems

We study the question of uniqueness of minimizers of the weighted least gradient problem min{∫ _Ω|Dv|_a : v∈ BV_loc(Ω∖ S), v|_∂Ω= f } , where ∫ _Ω|Dv|_a is the total variation with respect to the weight function a and S is the set of zeros of the function a . In contrast with previous results, which assume that the weight a∈ C^1,1(Ω ) and is bounded away from zero, here a is only assumed to be continuous, and is allowed to vanish and also be discontinuous in certain subsets of Ω . We assume instead existence of a C^1 minimizer. This problem arises naturally in the hybrid inverse problem of imaging electric conductivity from interior knowledge of the magnitude of one current density vector field, where existence of a C^1 minimizer is known a priori.