Superconducting dome in the LaAlO$_3$/SrTiO$_3$ interfaces as a direct effect of the extended s-wave symmetry of the gap

The two-dimensional electron gas (2DEG) at the LaAlO$_3$/SrTiO$_3$ interface exhibits gate tunable superconductivity with a domelike shape of $T_C$ as a function of electron concentration. Here, we propose that the experimentally observed behavior can be explained as a direct effect of the dominant $extended$ $s$-$wave$ symmetry of the superconducting gap. Our results quantitatively agree with the experimental data. Additionally, our calculations based on both the Hartee-Fock and the Guzwiller approximations show that the influence of electronic correlations on the paired state in the considered system is not that crucial as commonly believed.