An Lp-Comparison, $p\in (1,\infty )$, on the Finite Differences of a Discrete Harmonic Function at the Boundary of a Discrete Box

Abstract

AbstractIt is well-known that for a harmonic function u defined on the unit ball of the d-dimensional Euclidean space, d ≥ 2, the tangential and normal component of the gradient ∇u on the sphere are comparable by means of the Lp-norms, $p\in (1,\infty )$ p ∈ ( 1 , ∞ ) , up to multiplicative constants that depend only on d,p. This paper formulates and proves a discrete analogue of this result for discrete harmonic functions defined on a discrete box on the d-dimensional lattice with multiplicative constants that do not depend on the size of the box.