Relaxation and Integral Representation for Functionals of Linear Growth on Metric Measure spaces

Heikki Hakkarainen, Juha Kinnunen, Panu Lahti, Pekka Lehtelä

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9 Citations (Scopus)
168 Downloads (Pure)

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 defined via relaxation, and it defines 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.
Original languageEnglish
Article number13
Pages (from-to)288–313
JournalANALYSIS AND GEOMETRY IN METRIC SPACES
Volume2016
Issue number4
DOIs
Publication statusPublished - 2016
MoE publication typeA1 Journal article-refereed

Keywords

  • calculus of variations
  • functionals of linear growth
  • relaxation
  • functions of bounded variation
  • analysis on metric measure spaces

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