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On Some Properties of Relative Capacity and Thinness in Weighted Variable Exponent Sobolev Spaces

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Abstract

We define the weighted relative p(.)-capacity and discuss its properties in the space \(W_\vartheta ^{1,p(.)}({\mathbb{R}^n})\). Also, we investigate some properties of the weighted variable Sobolev capacity. It is shown that there is a relation between these two capacities. Moreover, we introduce the notion of thinness related to this newly defined relative capacity and prove an equivalence statement for this thinness.

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Correspondence to C. Unal.

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Unal, C., Aydin, I. On Some Properties of Relative Capacity and Thinness in Weighted Variable Exponent Sobolev Spaces. Anal Math 46, 147–167 (2020). https://doi.org/10.1007/s10476-020-0014-1

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Key words and phrases

  • weighted variable exponent Sobolev space
  • relative capacity
  • Sobolev capacity
  • thinness

Mathematics Subject Classification

  • primary 32U20
  • 31B15
  • secondary 46E35
  • 43A15