Rarity of ${\mathcal{C}}^{1,1}$ solutions to the complex Monge--Ampère equation on weakly pseudoconvex domains

Abstract

We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Ampère equation with ${\mathcal{C}}^{\infty }$-smooth data does not in general admit ${\mathcal{C}}^{1,1}$-smooth solutions. This working draft is a prelude to potential-theoretic solutions to some extension problems for mappings that were thought to rely on such ${\mathcal{C}}^{1,1}$-smooth solutions.

7 pages; this is a working draft; comments welcome!