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

A growth lemma for certain discrete symmetric Laplacians defined on a lattice Zd δ = δZd ⊂ Rd with spacing δ is proved. The lemma implies a De Giorgi theorem, that the harmonic functions for these Laplacians are equi-H¨older continuous, δ → 0. These results are then applied to establish regularity properties for the harmonic maps defined on Zd δ 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