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A Tour on p(x)-Laplacian Problems When p = ∞

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Mathematical Analysis, Approximation Theory and Their Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 111))

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Abstract

Most of the times, in problems where the p(x)-Laplacian is involved, the variable exponent p(⋅ ) is assumed to be bounded. The main reason for this is to be able to apply standard variational methods. The aim of this paper is to present the work that has been done so far, in problems where the variable exponent p(⋅ ) equals infinity in some part of the domain. In this case the infinity Laplace operator arises naturally and the notion of weak solution does not apply in the part where p(⋅ ) becomes infinite. Thus the notion of viscosity solution enters into the picture. We study both the Dirichlet and the Neumann case.

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Acknowledgement

The authors would like to thank the anonymous referee for his useful remarks and suggestions.

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Authors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens, 15780, Greece

    Yiannis Karagiorgos & Nikos Yannakakis

Authors
  1. Yiannis Karagiorgos
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  2. Nikos Yannakakis
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Correspondence to Nikos Yannakakis .

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Editors and Affiliations

  1. Department of Mathematics, National Technical Unversity of Athens, Athens, Greece

    Themistocles M. Rassias

  2. Department of Mathematics, Netaji Subhas Instit. of Tech., New Delhi, India

    Vijay Gupta

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Karagiorgos, Y., Yannakakis, N. (2016). A Tour on p(x)-Laplacian Problems When p = ∞. In: Rassias, T., Gupta, V. (eds) Mathematical Analysis, Approximation Theory and Their Applications. Springer Optimization and Its Applications, vol 111. Springer, Cham. https://doi.org/10.1007/978-3-319-31281-1_15

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