W-Sobolev spaces

Fix strictly increasing right continuous functions with left limits and periodic increments, W i : R → R , i = 1 , … , d , and let W ( x ) = ∑ i = 1 d W i ( x i ) for x ∈ R d . We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients ∇ W f = ( ∂ W 1 f , … , ∂ W d f ) . 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.