Least Gradient Problems with Neumann Boundary Condition

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.