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Operator equations and duality mappings in Sobolev spaces with variable exponents

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Abstract

After studying in a previous work the smoothness of the space

$U_{\Gamma _0 } = \{ u \in W^{1,p( \cdot )} (\Omega );u = 0on\Gamma _0 \subset \Gamma = \partial \Omega \} ,$

where dΓ − measΓ0 > 0, with p(·) ∈ \(\mathcal{C}(\bar \Omega )\) and p(x) > 1 for all x ∈ \(\bar \Omega \), the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space. The results obtained in this direction are used for proving existence results for operator equations having the form J φ u = N, where J φ is a duality mapping on \(U_{\Gamma _0 } \) corresponding to the gauge function φ, and N f is the Nemytskij operator generated by a Carathéodory function f satisfying an appropriate growth condition ensuring that N f may be viewed as acting from \(U_{\Gamma _0 } \) into its dual.

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Authors and Affiliations

  1. Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong, China

    Philippe G. Ciarlet

  2. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Street, 010014, Bucharest, Romania

    George Dinca

  3. Department of Mathematics and Computer Science, Technical University of Civil Engineering, 124 Lacul Tei Blvd., 020396, Bucharest, Romania

    Pavel Matei

Authors
  1. Philippe G. Ciarlet
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  2. George Dinca
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  3. Pavel Matei
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Correspondence to Philippe G. Ciarlet.

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Ciarlet, P.G., Dinca, G. & Matei, P. Operator equations and duality mappings in Sobolev spaces with variable exponents. Chin. Ann. Math. Ser. B 34, 639–666 (2013). https://doi.org/10.1007/s11401-013-0797-5

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  • DOI: https://doi.org/10.1007/s11401-013-0797-5

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