On the Regularity of Harmonic Functions and Spherical Harmonic Maps Defined on Lattices

Abstract

A growth lemma for certain discrete symmetric Laplacians defined on a lattice Zδd=δZd⊂Rd with spacing δ is proved. The lemma implies a De Giorgi theorem, that the harmonic functions for these Laplacians are equi-Hölder continuous, δ→0. These results are then applied to establish regularity properties for the harmonic maps defined on Zdδ and taking values in an n-dimensional sphere Sn, uniform in δ. Questions of the convergence δ→0 and the Dirichlet problem for these discrete harmonic maps are also addressed.