  We use first principles density functional theory calculations to investigate
the effect of biaxial strain in the low-temperature structural and
ferroelectric properties of [111]-oriented SrTiO$_3$. We find that [111]
biaxial strain, achievable by coherent epitaxial growth along the [111]
direction, induces structural distortions in SrTiO$_3$ that are not present in
either bulk or [001]-oriented SrTiO$_3$. Under [111] biaxial strain, SrTiO$_3$
displays ferroelectricity at tensile strain, and paraelectricity at compressive
strain. We compute the phonon spectrum and macroscopic polarization of
SrTiO$_3$ as a function of [111] biaxial strain, and relate our results to the
predictions of the free energy phenomenological model of Pertsev, Tagantsev and
Setter [Phys. Rev. B 61, 825 (2000); Phys. Rev. B 65, 219901 (2002)].
