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.
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.
We study relations between the variational Sobolev 1capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak (1, 1)-Poincaré inequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for 1 < p < ∞ to also cover the more geometric case p = 1. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.