## ABSTRACT

One of the reasons for the huge development of the theory of classical Lebesgue and Sobolev spaces Lp and W 1,p (where 1 ≤ p ≤ ∞) is the description of many phenomena arising in applied sciences. For instance, many materials can be modeled with sufficient accuracy using the function spaces Lp and W 1,p, where p is a fixed constant. For some nonhomogeneous materials, for instance electrorheological fluids (sometimes referred to as “smart fluids”), this approach is not adequate, but rather the exponent p should be allowed to vary. This leads us to the study of variable exponent Lebesgue and Sobolev spaces, Lp(x) and W 1,p(x), where p is a real-valued function.