  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, \[
BV_f(\Omega):=\{v\in BV(\Omega): \ \ \forall x\in \partial \Omega, \ \
\lim_{r\to 0} \ \esssup_{y\in \Omega, |x-y|<r} |f(x) - v(y)| = 0 \ \}
%BV_f(\Omega)=\{u\in BV(\Omega): {0.1cm} u|_{\partial \Omega}=f {0.1cm}
\hbox{and} {0.1cm} {0.1cm} u {0.1cm} \hbox{is continuous at} {0.1cm} \partial
\Omega \}. \] and $\varphi(x,\xi)$ is a function that, among other properties,
is convex and homogeneous of degree 1 with respect to the $\xi$ variable. In
particular we prove that if $a\in C^{1,1}(\Omega)$ is bounded away from zero,
then minimizers of the weighted least gradient problem $\inf_{u \in
BV_f}\int_{\Omega} a|Du|$ are unique in $BV_f(\Omega)$. We construct
counterexamples to show that the regularity assumption $a\in C^{1,1}$ is sharp,
in the sense that it can not be replaced by $a\in C^{1,\alpha}(\Omega)$ with
any $\alpha<1$.
