On any weakly pseudoconvex B-regular domain, the classical Dirichlet problem for the complex Monge-Ampère equation with C^∞ data does not in general admit C^{1,1} solutions.
Krantz and Song-Ying Li,On the existence of smooth plurisubharmonic solutions for certain de- generate Monge–Amp` ere equations, Complex Variables Theory Appl.41, no
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Rarity of $\boldsymbol{\mathcal{C}^{1,1}}$ solutions to the complex Monge--Amp\`ere equation on weakly pseudoconvex domains
On any weakly pseudoconvex B-regular domain, the classical Dirichlet problem for the complex Monge-Ampère equation with C^∞ data does not in general admit C^{1,1} solutions.