Weighted variable exponent Sobolev spaces and some applications

This thesis study consists of three basic sections. The first part of it has the characteristics of introduction, in this section the history of weighted variable exponent Lebesgue and Sobolev spaces and some applications. The basic definitions and theorems are given in second section. The notion of capacity has mentioned in the first part of findings section, in particular, relative -capacity is presented for an alternative of Sobolev -capacity in weighted variable exponent Sobolev spaces. In the second part, weighted variable exponent Sobolev spaces with zero boundary values are introduced and investigated their some basic properties. Also, the Poincaré inequality is proved in these spaces. In addition, a new capacity in is defined and given some inequalities between it and Sobolev -capacity. In the third part, -finely open sets defined by the relative -capacity are investigated. Moreover, the fine topology, which is generated these sets and is finer than Euclidean topology, is extented to weighted variable exponent case. In the fourth part, an intersection space between weighted classical Lebesgue spaces and weighted variable exponent Sobolev spaces has revealed with a suitable norm. Moreover, some properties and continuous-compact embeddings of this space are examined.