Least gradient problems with Neumann boundary condition

We study existence of minimizers of the least gradient probleminfv∈BVg⁡∫Ωφ(x,Dv), where BVg={v∈BV(Ω):∫∂Ωgv=1}, φ(x,p):Ω×Rn→R is a convex, continuous, and homogeneous function of degree 1 with respect to the p variable, and g satisfies the compatibility condition ∫∂ΩgdS=0. We prove that for every 0≢g∈L∞(∂Ω) there are infinitely many minimizers in BV(Ω). Moreover there exists a divergence free vector field T∈(L∞(Ω))n that determines the structure of level sets of all minimizers, i.e. T determines Du|Du|, |Du|-a.e. in Ω, 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.