Plurisubharmonic envelopes and supersolutions

We make a systematic study of (quasi-)plurisubharmonic envelopes on compact K"ahler manifolds, as well as on domains of ${\mathbb{C}}^n$, by using and extending an approximation process due to Berman [Ber13]. We show that the quasi-psh envelope of a viscosity super-solution is a pluripotential super-solution of a given complex Monge-Amp`ere equation. We use these ideas to solve complex Monge-Amp`ere equations by taking lower envelopes of super-solutions.