Potential Analysis

, 25:205 | Cite as

The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values

  • Petteri Harjulehto
  • Peter Hästö
  • Mika Koskenoja
  • Susanna Varonen


We define and study variable exponent Sobolev spaces with zero boundary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent.

Key words

variable exponent Sobolev space zero boundary values Sobolev capacity Poincaré inequality Dirichlet energy integral 

Mathematics Subject Classifications (2000)

46E35 31C45 35J65 


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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  • Petteri Harjulehto
    • 1
  • Peter Hästö
    • 2
  • Mika Koskenoja
    • 1
  • Susanna Varonen
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland
  2. 2.Department of Mathematical SciencesNTNUTrondheimNorway

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