Abstract:This article studies an integral representation of functionals of linear growth on metric measure spaces with a doubling measure and a Poincaré inequality. Such a functional is de ned via relaxation, and it de nes a Radon measure on the space. For the singular part of the functional, we get the expected integral representation with respect to the variation measure. A new feature is that in the representation for the absolutely continuous part, a constant appears already in the weighted Euclidean case. As an application we show that in a variational minimization problem involving the functional, boundary values can be presented as a penalty term.

In this paper we show existence of traces of functions of bounded variation on the boundary of a certain class of domains in metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality, and obtain L 1 estimates of the trace functions. In contrast with the treatment of traces given in other papers on this subject, the traces we consider do not require knowledge of the function in the exterior of the domain. We also establish a Maz'yatype inequality for functions of bounded variation that vanish on a set of positive capacity.

We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We propose a notion of domain with boundary of positive mean curvature and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here least gradient is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the boundary trace of the solution exists and agrees with the given boundary data. This extends the result of Sternberg, Williams and Ziemer [27] to the non-smooth setting. Via counterexamples we also show that uniqueness of solutions and existence of continuous solutions can fail, even in the weighted Euclidean setting with Lipschitz weights. * The authors thank Estibalitz Durand-Cartagena, Marie Snipes and Manuel Ritoré for fruitful discussions about the subject of the paper.

Abstract:In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modi cation on a set of measure zero) are continuous everywhere outside their jump sets. As a tool, we develop some stability properties of sequences of least gradient functions. We also apply these tools to prove a maximum principle for functions of least gradient that arise as solutions to a Dirichlet problem.

In this article we study minimizers of functionals of linear growth in metric measure spaces. We introduce the generalized problem in this setting, and prove existence and local boundedness of the minimizers. We give counterexamples to show that in general, minimizers are not continuous and can have jump discontinuities inside the domain.