  We consider a variational anti-plane lattice model and demonstrate that at
zero temperature, there exist locally stable states containing screw
dislocations, given conditions on the distance between the dislocations, and on
the distance between dislocations and the boundary of the crystal. In proving
our results, we introduce approximate solutions which are taken from the theory
of dislocations in linear elasticity, and use the inverse function theorem to
show that local minimisers lie near them. This gives credence to the commonly
held intuition that linear elasticity is essentially correct up to a few
spacings from the dislocation core.
