Embedding operators of Sobolev spaces with variable exponents and applications

Abstract

We introduce the vector-valued Sobolev spaces W-m,W-p(x) (Omega; E-0, E) with variable exponent associated with two Banach spaces E-0 and E. The most regular space E-alpha is found such that the differential operator D-alpha is bounded and compact from W-m,W-p(x)(Omega; E-0, E) to L-q(x)(Omega; E-alpha ), where E-alpha are interpolation spaces between E-0 and E is depending on alpha = (alpha(1), alpha(2),..., alpha(n)) and the positive integer m, where Omega subset of R-n is a region such that there exists a bounded linear extension operator from W-m,W-p(x) (Omega; E-0, E) to W-m,W-p(x) (R-n; E(A), E). The function p(x) is Lipschitz continuous on Omega and q(x) is a measurable function such that 1 < p(x) <= q(x) <= np(x)/n-mp(x) for a.e. x is an element of(Omega) over bar. Ehrling-Nirenberg-Gagilardo type sharp estimates for mixed derivatives are obtained. Then, by using this embedding result, the separability properties of abstract differential equations are established.