Basic Properties of Variable Sobolev Spaces

Abstract

In this chapter we present the elementary theory of variable Sobolev spaces. Unlike Chap. 2, where we systematically developed a complete theory of variable Lebesgue spaces, our goal here is less ambitious. Our aim is to illustrate how the theorems and techniques given in previous chapters can be applied to the variable Sobolev spaces. Consequently, many of our results are not given in the fullest generality possible, and there are a number of results from the classical theory of Sobolev spaces that we do not discuss.