Embedding operators of Sobolev spaces with variable exponents and applications

We introduce the vector-valued Sobolev spaces W m , p ( x ) (Ω; E 0 , E ) with variable exponent associated with two Banach spaces E 0 and E . The most regular space E α is found such that the differential operator D α is bounded and compact from W m , p ( x ) (Ω; E 0 , E ) to L q ( x ) (Ω; E α ), where E α are interpolation spaces between E 0 and E is depending on α = ( α 1 , α 2 ,..., α n ) and the positive integer m , where Ω ⊂ ℝ n is a region such that there exists a bounded linear extension operator from W m , p ( x ) (Ω; E 0 , E ) to W m , p ( x ) (ℝ n ; E ( A ), E ). The function p ( x ) is Lipschitz continuous on Ω and q ( x ) is a measurable function such that for a.e. . 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.