Uniqueness of minimizers of weighted least gradient problems arising in conductivity imaging

We prove uniqueness for minimizers of the weighted least gradient problem $$\inf \{{\int }_{\mathrm{\Omega }}a\mathrm{\mid }Du\mathrm{\mid }:u\in BV(\mathrm{\Omega }),u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=f\}\mathrm{.}$$ The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $\mathrm{\Omega }$. Existence is assumed a priori. Our approach is motivated by the hybrid inverse problem of imaging electric conductivity from interior knowledge (obtainable by MRI) of the magnitude of one current density vector field.