  We study the modification of the electronic structure in the strong
spin-orbit coupled Sr$_2$IrO$_4$ by epitaxial strain using density functional
methods. Structural optimization shows that strain changes the internal
structural parameters such as the Ir-O-Ir bond angle, which has an important
effect on the band structure. An interesting prediction is the $\Gamma - $X
crossover of the valence band maximum with strain, while the conduction minimum
at M remains unchanged. This in turn suggests strong strain dependence of the
transport properties for the hole doped system, but not when the system is
electron-doped. Taking the measured value of the $\Gamma-X$ separation for the
unstrained case, we predict the $\Gamma - $X crossover of the valence band
maximum to occur for the tensile epitaxial strain $e_{xx} \approx 3\%$. A
minimal tight-binding model within the $J_{\rm eff} = 1/2$ subspace is
developed to describe the main features of the band structure. The optical
absorption spectra under epitaxial strain are computed using density-functional
theory, which explains the observed anisotropy in the optical spectra with the
polarization of the incident light. We show that the optical transitions
between the Ir (d) states, which are dipole forbidden, can be explained in
terms of the admixture of Ir (p) orbitals with the Ir (d) bands.
