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

We prove uniqueness for minimizers of the weighted least gradient problem $$\inf \left\lbrace \int_{\Omega} a|Du|: \ \ u\in BV(\Omega), \ \ u|_{\partial \Omega}=f \right\rbrace.$$ The weight function $a$ is assumed to be continuous and it is allowed to vanish in certain subsets of $\Omega$. 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.