Open Problems in Pluripotential Theory — Question 24

Definitions and mathematical statement

Let $n\geq 1$ be an integer and let $\Omega\subset{\mathbb{C}}^n$ be a bounded domain with smooth strictly pseudoconvex boundary. Here strict pseudoconvexity means that a smooth local defining function $\rho$, with $\Omega=\{\rho<0\}$ and ${\,\mathrm{d}}\rho\ne0$ on the boundary, has positive Levi form on every nonzero complex tangent vector to $\partial\Omega$. Let $\phi\in\mathcal C^{1,1}(\partial\Omega;{\mathbb{R}})$, meaning that in smooth boundary coordinates its first derivatives are locally Lipschitz.

A function is plurisubharmonic if it is upper semicontinuous and its restriction to each complex line is subharmonic wherever defined. Use the convention $d^c=\frac{i}{2}(\bar\partial-\partial)$, so that $dd^c=i\partial\bar\partial$. For a locally bounded plurisubharmonic function, interpret $(dd^c u)^n$ as its Bedford--Taylor complex Monge--Ampère measure. The normalization of $d^c$ does not affect the equation that this measure vanishes.

Let $u\in\operatorname{PSH}(\Omega)\cap\mathcal C(\overline\Omega;{\mathbb{R}})$ be the unique solution specified in the source of \[ \begin{cases} (dd^c u)^n=0 & \text{in }\Omega,\\ u=\phi & \text{on }\partial\Omega. \end{cases} \] Requested statement (attributed to Demailly). For every such $\Omega$ and $\phi$, prove, using complex analytic methods, that $u$ is $\mathcal C^{1,1}$-smooth in $\Omega$.

The interior regularity conclusion is interpreted here as $u\in\mathcal C^{1,1}_{\mathrm{loc}}(\Omega)$: in the real coordinates of ${\mathbb{C}}^n\cong{\mathbb{R}}^{2n}$, the function is continuously differentiable and its first derivatives are locally Lipschitz. The supplied source writes $\mathcal C^{1,1}$ in $\Omega$ without specifying whether a uniform bound throughout $\Omega$ is intended. That stronger interpretation remains unresolved; no regularity on $\overline\Omega$ is asserted here. The requirement to use complex analytic methods is part of the problem, but the source does not define the permitted class of methods.

Short English statement

For the continuous plurisubharmonic solution of the homogeneous complex Monge--Ampère Dirichlet problem on a smooth bounded strictly pseudoconvex domain, with $\mathcal C^{1,1}$ boundary data, give a proof by complex analytic methods of interior $\mathcal C^{1,1}$ regularity. The intended strength of the interior bounds and the permitted methods require clarification.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $800893$, exact source problem code catalogue entry, title Open Problems in Pluripotential Theory --- Question 24. The record labels the problem open; this is supplied registry metadata, not an independently verified mathematical or literature assessment.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 24. Supplied source URL: https://arxiv.org/abs/1511.00705. The question itself is attributed to Demailly by the supplied statement.
• Supplied provenance: source accessed on 2026-10-16; extraction method arxiv_latex. This access date and provenance are supplied curation information, not a claim of independent source consultation. Supplied upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.