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 Z(delta)(d) = deltaZ(d) subset of R-d with spacing delta is proved. The lemma implies a De Giorgi theorem rem, that the harmonic functions for these Laplacians are equi-Holder continuous, delta --> 0. These results are then applied to establish regularity properties for the harmonic maps defined on Z(delta)(d) and taking values in an n-dimensional sphere S', uniform in delta. Questions of the convergence delta --> 0 and the Dirichlet problem for these discrete harmonic maps are also addressed. (C) 2001 Academic Press.