On some properties of relative capacity and thinness in weighted variable exponent Sobolev spaces

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