Estimates On the Variance of Some Homogenization Problems

We use `p estimates together with Brascamp-Lieb inequalities to obtain bounds on the variance of the solution u" to the elliptic equation r" a(x="; ')r"u" = r" f . The variance is shown to be O("r) where the exponent r depends on the contrast of a(x; '). In two dimensions, we show that variance of the Greens function of the above elliptic operator is bounded. We also obtain decay rates for the derivative of the Greens function. Except for this later estimate, all estimates assume that the underlying probability space is induced by a convex Gibbs distribution. Mathematics Subject Classification: 35R60, 82D30, 60J75, 60J60