Abstract

We revisit the question of existence and regularity of minimizers to the weighted least gradient problem with Dirichlet boundary condition inf {integral(Omega) a(x)vertical bar Du vertical bar : u is an element of BV(Omega), u vertical bar(partial derivative Omega) = g}, where g is an element of C(partial derivative Omega), and a is an element of C-2 ((Omega) over bar) is a weight function that is bounded away from zero. Under suitable geometric conditions on the domain Omega subset of R-n, 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 a(2/(n-1))I(n). 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). (C) 2018 Elsevier Ltd. All rights reserved.