A note on the discrete Aleksandrov-Bakelman maximum principle

In previous works, we have established discrete versions of the Aleksandrov -Bakelman maximum principle for elliptic operators, on general meshes in E uclidean space. In this paper, we prove a variant of these estimates in ter ms of a discrete analogue of the determinant of the coefficient matrix in t he differential operator case. Our treatment depends on an interesting conn ection between the determinant and volumes of cells in the underlying mesh.