The Dirichlet Problem for Complex Monge-ampère Equations and Regularity of the Pluri-complex Green Function

(1.1) det(uzj z̄k) = ψ(z, u,∇u) in Ω, u = φ on ∂Ω and related questions. When Ω is a strongly pseudoconvex domain, this problem has received extensive study. In [4]-[6], E. Bedford and B. A. Taylor established the existence, uniqueness and global Lipschitz regularity of generalized pluri-subharmonic solutions. S.-Y. Cheng and S.-T. Yau [8], in their work on complete Kähler-Einstein metrics on non-compact complex manifolds, solved (1.1) for ψ = e and φ = +∞, obtaining a solution in C∞(Ω). In 1985, L. Caffarelli, J. J. Kohn, L. Nirenberg and J. Spruck [7] proved the existence of classical pluri-subharmonic solutions of (1.1) for the non-degenerate case ψ > 0, under suitable conditions on ψ. The degenerate case ψ ≥ 0 has also attracted a lot of attention, and counterexamples have been found showing that there need not be a C solution (see [3], [11]). It is of interest in complex analysis to ask whether C regularity holds for the degenerate case; see [1] for related results and further references. In [20], S.-Y. Li studied the Neumann problem for complex Monge-Ampère equations. In this paper we treat the Dirichlet problem (1.1) for general domains which are not necessarily pseudoconvex. We shall prove