W-Sobolev spaces

Fix strictly increasing right continuous functions with left limits and periodic increments, Wi:R→R, i=1,…,d, and let for x∈Rd. We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients ∇Wf=(∂W1f,…,∂Wdf). Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W-generalized elliptic equations, and uniqueness results for W-generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.