  We construct a domain wall solution in $F(R)$ gravity. We reconstruct a
static domain wall solution in a scalar field theory. We also reconstruct an
explicit $F(R)$ gravity model in which a static domain wall solution can be
realized. Moreover, we show that there could exist an effective (gravitational)
domain wall in the framework of $F(R)$ gravity. In addition, it is demonstrated
that a logarithmic non-minimal gravitational coupling of the electromagnetic
theory in $F(R)$ gravity may produce time-variation of the fine structure
constant which may increase with decrease of the curvature, and that this model
would be ruled out by the constraints on the time variation of the fine
structure constant from quasar absorption lines. We also present cosmological
consequences of the coupling of the electromagnetic field to a scalar field as
well as the scalar curvature and discuss the relation between variation of the
fine structure constant and the breaking of the conformal invariance of the
electromagnetic field.
