Generalized local maximum principles for finite-difference operators

Abstract:

The generalized local maximum principle for a difference operator ${L_h}$ asserts that if ${L_h}u(x) > 0$ then $\Gamma u$ cannot attain its positive maximum at the net-point x. Here $\Gamma$ is a local net-operator such that $\Gamma u = u + O(h)$ for any smooth function u. This principle, with simple forms of $\Gamma$, is proved for some quite general classes of second-order elliptic operators ${L_h}$, whose associated global matrices are not necessarily monotone. It is shown that these generalized principles can be used for easy derivation of global a priori estimates to the solutions of elliptic difference equations and to their difference-quotients. Some examples of parabolic difference equations are also treated.