  We study the zero temperature phase diagram of Ising spin systems in two
dimensions in the presence of competing interactions, long range
antiferromagnetic and nearest neighbor ferromagnetic of strength J. We first
introduce the notion of a "corner energy" which shows, when the
antiferromagnetic interaction decays faster than the fourth power of the
distance, that a striped state is favored with respect to a checkerboard state
when J is close to J_c, the transition to the ferromagnetic state, i.e., when
the length scales of the uniformly magnetized domains become large. Next, we
perform detailed analytic computations on the energies of the striped and
checkerboard states in the cases of antiferromagnetic interactions with
exponential decay and with power law decay r^{-p}, p>2, that depend on the
Manhattan distance instead of the Euclidean distance. We prove that the striped
phase is always favored compared to the checkerboard phase when the scale of
the ground state structure is very large. This happens for J\lesssim J_c if
p>3, and for J sufficiently large if 2<p<=3. Many of our considerations
involving rigorous bounds carry over to dimensions greater than two and to more
general short-range ferromagnetic interactions.
