Linear elliptic difference inequalities with random coefficients

Abstract

Abstract.We prove various pointwise estimates for solutions of linear elliptic difference inequalities with random coefficients. These estimates include discrete versions of the maximum principle of Aleksandrov and Harnack inequalities and Holder estimates of Krylov and Safonov for elliptic differential operators with bounded coefficients.

On the discrete maximum principle for parabolic difference operators

“…In this paper we establish discrete versions of the Krylov maximum principle [4,5] established in our previous work [6] (Theorem 2.1). As in [6], our estimâtes are formulated in such a way that their continuous versions follow via Taylor's formula.…”

“…The spatial part of our différence operators will be determined by second order différence operators of the form L h u{x, t) = J^ a(x, t, y) Ôy u(Xj t) + y C O (1.5) with real coefficients a, b, c, having compact support with respect to y, and satisfying as in [6], the condition a(x, t, j ) -| \y\ \b(x,t,y)\ 0 .…”

“…Consequently only (1.6) and (1.8) will be essential for us with (1.9) being replaced by a weaker condition (see (2.6)). We shall formulate a discrete version of the Krylov maximum principle in the next section for operators 3£ ^ satisfying stricter conditions than (1.6) and (1.8) corresponding to the non-degeneracy condition assumed in [6]. In Section 3 we provide some basic inequalities from Krylov's paper [4] which are used in our proof which is supplied in Section 4.…”

“…Finally in Section 5 we relate the discrete and continuous versions of the maximum principle. In an ensuing paper, we shall apply Theorem2.1 to the dérivation of local estimâtes, corresponding to those in the elliptic case [6].…”

“…As in [6], our estimâtes are formulated in such a way that their continuous versions follow via Taylor's formula. Our discretizations of the operator (1.1) will involve linear différence operators, essentially of positive type, acting on space-time mesh functions.…”