The singular Dirichlet problem for the complex Monge-Ampère operator on complex manifolds (Q914046)

Let M be an n-dimensional Stein manifold with volume form dV and \(\Omega\) a relatively compact strictly pseudoconvex domain in M without any assumption of boundary regularity. If P(\(\Omega\)) denotes the cone of plurisubharmonic functions on \(\Omega\), according to Bedford-Taylor (1982) \((dd^ cu)^ n:=\bigwedge^ ndd^ cu\) is well defined as a positive Radon measure for \(u\in P(\Omega)\cap L^{\infty}_{loc}(\Omega).\) The following theorem is proved. Theorem: Let \(F\in L^{\infty}_{loc}(R\times \Omega)\) be a positive function such that F(t,z) is bounded on \(\Omega\) for each \(t\in R\) fixed, and continuous, nondecreasing in t for every fixed \(z\in \Omega\). Suppose that \(F(t,z)\geq e^{kt}\) in some neighborhood of each (t,z)\(\in R\times \Omega\) for some constant \(k>0\). Then the upper envelope, \[ u(z):=\sup \{v(z);\quad v\in P(\Omega)\cap L^{\infty}_{loc}(\Omega),\quad (dd^ cv)^ n\geq F(v,z)dV\} \] is a maximal generalized solution of the Dirichlet problem: \[ u\in P(\Omega)\cap L^{\infty}_{loc}(\Omega),\quad \limsup_{Q\ni z\to \zeta}u(z)=+\infty \text{ for } \zeta \in \partial \Omega,\quad (dd^ cu)^ n=F(u,z)dV\text{ on } \Omega. \]