Continuity of minimizers to weighted least gradient problems

Abstract

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.