  We study existence of minimizers of the least gradient problem \[\inf_{v \in
BV_g} \int_{\Omega}\varphi(x, Dv),\] where $BV_g=\{v \in BV(\Omega):
\int_{\partial \Omega}gv=1\}$, $\varphi(x,p): \Omega\times \R^n \rightarrow \R$
is a convex, continuous, and homogeneous function of degree $1$ with respect to
the $p$ variable, and $g$ satisfies the comparability condition $\int_{\partial
\Omega} g dS=0$. We prove that for every $0\not \equiv g \in
L^{\infty}(\partial \Omega)$ there are infinitely many minimizers in
$BV(\Omega)$. Moreover there exists a divergence free vector field $T\in
(L^{\infty}(\Omega))^n$ that determines the structure of level sets of all
minimizers, i.e. $T$ determines $\frac{Du}{|Du|}$, $|Du|-$ a.e. in $\Omega$,
for every minimizer $u$. We also prove some existence results for general
1-Laplacian type equations with Neumann boundary condition. A numerical
algorithm is presented that simultaneously finds $T$ and a minimizer of the
above least gradient problem. Applications of the results in conductivity
imaging are discussed.
