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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 continuousand it is allowed to vanish in certain subsets of $\Omega$. Existence isassumed a priori. Our approach is motivated by the hybrid inverse problem ofimaging electric conductivity from interior knowledge (obtainable by MRI) ofthe magnitude of one current density vector field.

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