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manuscripta mathematica

, Volume 121, Issue 3, pp 339–366 | Cite as

Moser iteration for (quasi)minimizers on metric spaces

  • Anders Björn
  • Niko MarolaEmail author
Article

Abstract

We study regularity properties of quasiminimizers of the p-Dirichlet integral on metric measure spaces. We adapt the Moser iteration technique to this setting and show that it can be applied without an underlying differential equation. However, we have been able to run the Moser iteration fully only for minimizers. We prove Caccioppoli inequalities and local boundedness properties for quasisub- and quasisuperminimizers. This is done in metric spaces equipped with a doubling measure and supporting a weak (1, p)-Poincaré inequality. The metric space is not required to be complete. We also provide an example which shows that the dilation constant from the weak Poincaré inequality is essential in the condition on the balls in the Harnack inequality. This fact seems to have been overlooked in the earlier literature on nonlinear potential theory on metric spaces.

Mathematics Subject Classification (2000)

Primary 49N60 Secondary 35J60 Secondary 49J27 

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Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of MathematicsLinköpings UniversitetLinköpingSweden
  2. 2.Institute of MathematicsHelsinki University of TechnologyHelsinkiFinland

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