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

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

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