Sobolev capacity of the space \(W^{1,p(\cdot)} (\mathbb{R}^n)\)
The goal of the authors is to define the notion of Sobolev capacity on the generalized Sobolev space \(W^{1,p(\cdot)}(\mathbb{R}^n)\) and study its properties. They show that the Sobolev \(p(\cdot)\)-capacity is an outer measure and as a corollary they obtain that it is the Choquet capacity. They present several results that illustrate the utility of the Sobolev capacity. Moreover, the authors show that the capacity is related to the Hausdorff dimension. Using quasicontinuous representative in \(W^{1,p(\cdot)}(\mathbb{R}^n)\), the autors derive an estimate of the capacity of level sets of variable exponent Sobolev functions.
scientific article; zbMATH DE number 2111351