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

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Abstract

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.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, FIN-00014, Helsinki, Finland

    Petteri Harjulehto, Mika Koskenoja & Susanna Varonen

  2. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Peter Hästö

Authors
  1. Petteri Harjulehto
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  2. Peter Hästö
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  3. Mika Koskenoja
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  4. Susanna Varonen
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Correspondence to Petteri Harjulehto.

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Harjulehto, P., Hästö, P., Koskenoja, M. et al. The Dirichlet Energy Integral and Variable Exponent Sobolev Spaces with Zero Boundary Values. Potential Anal 25, 205–222 (2006). https://doi.org/10.1007/s11118-006-9023-3

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