Continuity of minimizers to weighted least gradient problems

We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary condition inf∫Ωa(x)|Du|:u∈BV(Ω),u|∂Ω=g,where g∈C(∂Ω), and a∈C2(Ω̄) is a weight function that is bounded away from zero. Under suitable geometric conditions on the domain Ω⊂Rn, we construct continuous solutions of the above problem for any dimension n≥2, by extending the Sternberg-Williams-Ziemer technique (Sternberg et al., 1992) to this setting of inhomogeneous variations. We show that the level sets of the constructed minimizer are minimal surfaces in the conformal metric a2∕(n−1)In. This result complements the approach in Jerrard et al. (2018) since it provides a continuous solution even in high dimensions where the possibility exists for level sets to develop singularities. The proof relies on an application of a strict maximum principle for sets with area minimizing boundary established by Leon Simon in Simon (1987).