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Stability and Continuity of Functions of Least Gradient

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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.


  • Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland
  • Department of Mathematics and Statistics, P.O. Box 68, FI-00014 University of Helsinki, Finland
  • Aalto University, School of Science and Technology, Department of Mathematics, P.O. Box 11100, FI-00076 Aalto, Finland
  • Department of Mathematical Sciences, P.O.Box 210025, University of Cincinnati, Cincinnati, OH 452210025, U.S.A.


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