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

TL;DR

This paper develops $W$-Sobolev spaces based on functions with jumps, establishes their properties, and applies them to analyze generalized PDEs, homogenization, and hydrodynamic limits in random environments.

Contribution

It introduces the $W$-Sobolev spaces, extending classical Sobolev theory to functions with jumps, and applies this framework to PDEs, homogenization, and hydrodynamic limits.

Findings

Established properties of $W$-Sobolev spaces analogous to classical Sobolev spaces.

Proved existence and uniqueness of solutions for $W$-generalized elliptic and parabolic equations.

Derived homogenization results and proved hydrodynamic limits for gradient processes in random environments.

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.