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Equivalence of viscosity and weak solutions for the normalized p(x)-Laplacian

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Abstract

We show that viscosity solutions to the normalized p(x)-Laplace equation coincide with distributional weak solutions to the strong p(x)-Laplace equation when p is Lipschitz and \(\inf p>1\). This yields \(\smash {C^{1,\alpha }}\) regularity for the viscosity solutions of the normalized p(x)-Laplace equation. As an additional application, we prove a Radó-type removability theorem.

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Authors and Affiliations

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

    Jarkko Siltakoski

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  1. Jarkko Siltakoski
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Correspondence to Jarkko Siltakoski.

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Communicated by Y. Giga.

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Siltakoski, J. Equivalence of viscosity and weak solutions for the normalized p(x)-Laplacian. Calc. Var. 57, 95 (2018). https://doi.org/10.1007/s00526-018-1375-1

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