Operator equations and duality mappings in Sobolev spaces with variable exponents

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.

Keywords

Monotone operators / Smoothness / Strict convexity / Uniform convexity / Duality mappings / Sobolev spaces with a variable exponent / Nemytskij operators

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