A Viscosity Approach to the Dirichlet Problem for Complex Monge-Amp\`ere Equations

The Dirichlet problem for complex Monge-Amp'ere equations with continuous data is considered. In particular, a notion of viscosity solutions is introduced; a comparison principle and a solvability theorem are proved; the equivalence between viscosity and pluripotential solutions is established; and an ABP-type of $L^{\mathrm{\infty }}$-estimate is achieved.