Calculus of Variations and Geometric Measure Theory

N. Shanmugalingam - P. Lahti - R. Korte - H. Hakkarainen

Stability and continuity of functions of least gradient

created by shanmugal on 08 Oct 2014
modified on 24 Jul 2015



Inserted: 8 oct 2014
Last Updated: 24 jul 2015

Journal: Analysis and Geometry in Metric Spaces
Volume: 3
Pages: 123-139
Year: 2015
Doi: 10.1515/agms-2015-0009


In this note we prove that on metric measure spaces, functions of least gradient, as well as local minimizers of the area functional (after modification 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.