Stability and Continuity of Functions of Least Gradient

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 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.

Keywords

• least gradient
• bv; metric measure spac
• approximate continuity
• continuity
• stability
• jump set
• dirichlet problem
• minimal surface
• primary 26b30
• secondary 31e99
• 31c45
• 26b15