  Exact ground states of two-dimensional Ising spin glasses with Gaussian and
bimodal (+- J) distributions of the disorder are calculated using a
``matching'' algorithm, which allows large system sizes of up to N=480^2 spins
to be investigated. We study domain walls induced by two rather different types
of boundary-condition changes, and, in each case, analyze the system-size
dependence of an appropriately defined ``defect energy'', which we denote by
DE. For Gaussian disorder, we find a power-law behavior DE ~ L^\theta, with
\theta=-0.266(2) and \theta=-0.282(2) for the two types of boundary condition
changes. These results are in reasonable agreement with each other, allowing
for small systematic effects. They also agree well with earlier work on smaller
sizes. The negative value indicates that two dimensions is below the lower
critical dimension d_c. For the +-J model, we obtain a different result, namely
the domain-wall energy saturates at a nonzero value for L\to \infty, so \theta
= 0, indicating that the lower critical dimension for the +-J model exactly
d_c=2.
