2009/11/23 by Alexandre B. Simas, Simas, Alexandre B., Fábio Valentim +1
Computer Science · Mathematics · Engineering · #Advanced Mathematical Modeling in Engineering #Nonlinear Partial Differential Equations #Advanced Numerical Methods in Computational Mathematics
paper · pdf · doi:10.48550/arxiv.0911.4177
Fix strictly increasing right continuous functions with left limits Wi:\bb R → \bb R, i=1,...,d, and let W(x) = ∑i=1d Wi(xi) for x∈\bb Rd. We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients ∇W f = (∂W1 f,...,∂Wd 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.