  We present results of self-consistent, high-resolution cosmological
simulations of galaxy formation at z~3. The simulations employ recently
developed recipe for star formation based on the local abundance of molecular
hydrogen, which is tracked self-consistently during the course of simulation.
The phenomenological H2 formation model accounts for the effects of
dissociating UV radiation of stars in each galaxy, as well as self-shielding
and shielding of H2 by dust, and therefore allows us to explore effects of
lower metallicities and higher UV fluxes prevalent in high redshift galaxies on
their star formation. We compare stellar masses, metallicities, and star
formation rates of the simulated galaxies to available observations of the
Lyman Break Galaxies (LBGs) and find a reasonable agreement. We find that the
Kennicutt-Schmidt (KS) relation exhibited by our simulated galaxies at z~3 is
substantially steeper and has a lower amplitude than the z=0 relation at
Sigma_gas < 100 Msun/pc^2. The predicted relation, however, is consistent with
existing observational constraints for the z~3 Damped Lyman $\alpha$ (DLA) and
LBGs. Our tests show that the main reason for the difference from the local KS
relation is lower metallicity of the ISM in high redshift galaxies. We discuss
several implications of the metallicity-dependence of the KS relation for
galaxy evolution and interpretation of observations.
