$W$-Sobolev spaces: Theory, Homogenization and Applications

Abstract

Fix strictly increasing right continuous functions with left limits $W_i:\bb R \to \bb R$, $i=1,...,d$, and let $W(x)={\sum }_{i=1}^d{W}_i(x_i)$ for $x\in\bb R^d$. We construct the $W$-Sobolev spaces, which consist of functions $f$ having weak generalized gradients ${\nabla }_{W}f=({\partial }_{W_1}f,...,{\partial }_{W_d}f)$. Several properties, that are analogous to classical results on Sobolev spaces, are obtained. $W$-generalized elliptic and parabolic equations are also established, along with results on existence and uniqueness of weak solutions of such equations. Homogenization results of suitable random operators are investigated. Finally, as an application of all the theory developed, we prove a hydrodynamic limit for gradient processes with conductances (induced by $W$) in random environments.

References in corpus (3)

• Hydrodynamic behavior of one dimensional subdiffusive exclusion processes with random conductances
• Hydrodynamic limit of gradient exclusion processes with conductances on $\bb Z^d$
• Random walks and exclusion processes among random conductances on random infinite clusters: homogenization and hydrodynamic limit