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∈BV_f(Ω)∫_Ωφ(x,Du), where f:∂Ω→R is continuous, and φ(x,ξ) is a function that, among other properties, is convex and homogeneous of degree 1 with respect to the ξ variable. In particular we prove that if a∈C_1,1(Ω) is bounded away from zero, then minimizers of the weighted least gradient problem inf_u∈BVf∫_Ωa∣Du∣ are unique in BVf(Ω). We construct counterexamples to show that the regularity assumption a∈C1,1 is sharp, in the sense that it can not be replaced by a∈C^1,α(Ω) with any α<1.a∈C1,α(Ω) with any α<1. [ABSTRACT FROM AUTHOR]