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Nonlinear parabolic capacity and polar sets of superparabolic functions

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Abstract

We extend the theory of the thermal capacity for the heat equation to nonlinear parabolic equations of the \(p\)-Laplacian type. We study definitions and properties of the nonlinear parabolic capacity and show that the capacity of a compact set can be represented via a capacitary potential. As an application, we show that polar sets of superparabolic functions are of zero capacity. The main technical tools used include estimates for equations with measure data and obstacle problems.

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Authors and Affiliations

  1. Department of Mathematics, Aalto University, P. O. Box 11100, 00076, Aalto, Finland

    Juha Kinnunen & Tuomo Kuusi

  2. Department of Mathematics and Statistics, University of Helsinki, P. O. Box 68, 00014, Helsinki, Finland

    Riikka Korte

  3. Department of Mathematics and Statistics, University of Jyväskylä, P. O. Box 35, 40014, Jyväskylä, Finland

    Mikko Parviainen

Authors
  1. Juha Kinnunen
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  2. Riikka Korte
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  3. Tuomo Kuusi
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  4. Mikko Parviainen
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Correspondence to Juha Kinnunen.

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Kinnunen, J., Korte, R., Kuusi, T. et al. Nonlinear parabolic capacity and polar sets of superparabolic functions. Math. Ann. 355, 1349–1381 (2013). https://doi.org/10.1007/s00208-012-0825-x

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  • DOI: https://doi.org/10.1007/s00208-012-0825-x

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