TITLE: Bedford-Taylor theory 
QUESTION [5 upvotes]: The Dirichlet problem for the Complex Monge-Ampere equation on a bounded pseudoconvex domain in $\mathbb{C}^n$ was studied in Bedford-Taylor's seminal paper wherein they defined $(dd^{c} u)^n$ for locally bounded plurisubharmonic $u$. But, they don't seem to use it in their Perron-type method, instead using a convex-measure-theoretic construction claiming that the upper envelope is not well-behaved. Now that we know more about psh functions, have people studied the Dirichlet problem without using the measure-theoretic construction of Goffman and Serrin? (and reproved Bedford-Taylor's results)

REPLY [4 votes]: In the book Degenerate Complex Monge-Ampère Equations by V. Guedj, A. Zeriahi, they circumvent the construction of Goffman and Serrin using ideas from viscosity theory.  Let $H_n^+$ denote the set of all semi-positive Hermitian $n$ by $n$ matrices and let $\dot{H}_n^+ \subset H_n^+$ denote the matrices with determinant $n^{-n}$. For $H \in  \dot{H}_n^+$ they define
$$ \Delta_H := \sum_{i,j=1}^n h_{i,j}\frac{\partial^2}{\partial z_i \partial \bar{z}_j}$$
and show that for $u \in PSH(\Omega) \cap L^\infty(\Omega)$, $0 \leq f \in C^0(\Omega)$,
$$ (dd^cu)^n \geq f \beta^n \iff \Delta_H u \geq f^{1/n} \text{ for all $H \in \dot{H}_n^+$.} $$
The envelopes are then constructed with regards to the second statement.