The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function

Cauchy–Riemann meet Monge–Ampère

“…Since Harrington [61] generalized this DiederichFornaess result to pseudoconvex domains with Lipschitz boundary, the conjecture also holds in this case. Further regularity of the pluricomplex Green function was established in [56] and [18] (see also [19]): if is C 2,1 -smooth and strongly pseudoconvex then for a fixed w ∈ we have G (·, w) ∈ C 1,1 (¯ \{w}). This is the highest regularity we can expect, Bedford and Demailly showed that G (·, w) does not have to be C 2 -smooth up to the boundary even if is C ∞ -smooth and strongly pseudoconvex.…”

A correction to “The Dirichlet problem for complex Monge–Ampère equations and regularity of the pluri-complex Green function”

“…It also affects the proof of Theorem 1.2 which states that the pluri-complex Green function g for a strongly pseudoconvex domain Q C C n with a logarithmic pole at a point £ G Q belongs to C 1 ' a (Jl -{£}) for any 0 < a < 1. Here we present a proof of this result independent of Lemma 3.1 of [1].…”

“…As a result, the proof of Theorem 1.3 in [1] is incomplete. It also affects the proof of Theorem 1.2 which states that the pluri-complex Green function g for a strongly pseudoconvex domain Q C C n with a logarithmic pole at a point £ G Q belongs to C 1 ' a (Jl -{£}) for any 0 < a < 1.…”

“…Bo GUAN I recently learned that the proof of Lemma 3.1 in [1] contains an error. As a result, the proof of Theorem 1.3 in [1] is incomplete.…”

Monge-Ampère Equations, Geodesics and Geometric Invariant Theory

“…Barrier methods for Monge-Ampère boundary value prolem have been developed by a number of authors, including Caffarelli, Kohn, Nirenberg, Spruck [5], [6], [7], and Guan [17]. Using such a barrier method, together with Yau's estimates in [30], X.X.…”