Existence and uniqueness of minimizers of general least gradient problems

Motivated by problems arising in conductivity imaging, we prove existence, uniqueness, and comparison theorems - under certain sharp conditions - for minimizers of the general least gradient problem \[\inf_{u\in BV_f(\Omega)} \int_{\Omega}\varphi(x,Du),\] where $f:\partial \Omega\to \R$ is continuous, \[ BV_f(\Omega):=\{v\in BV(\Omega): \ \ \forall x\in \partial \Omega, \ \ \lim_{r\to 0} \ \esssup_{y\in \Omega, |x-y|