  We revisit the question of existence and regularity of minimizers to weighted
least gradient problems on a fixed bounded domain, subject to a Dirichlet
boundary condition, in the case where the boundary data is continuous and the
weight function is C^2 and bounded away from zero. Under suitable geometric
conditions on the domain in R^n we construct continuous solutions of the above
variational problem in any dimension n>=2, by extending the
Sternberg-Williams-Ziemer technique to this setting of inhomogeneous
variations. We show that the level sets of the constructed minimizer are
minimal surfaces in a conformal metric determined by the weight function. This
results complements the approach of Jerrard, Moradifam and Nachman 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.
