Open Problems in Pluripotential Theory — Question 13

Definitions and mathematical statement

Let $n\geq 1$, and let $\Omega\subset{\mathbb{C}}^n$ denote the intended domain. A function is plurisubharmonic (psh) if it is upper semicontinuous and its restriction to each complex line is subharmonic wherever defined. For a continuous psh function $v$, write $(dd^c v)^n$ for its complex Monge--Ampère measure in the locally bounded sense, using one consistent normalization of $d^c$. For nonnegative Borel measures, $\mu\leq\nu$ means $\mu(E)\leq\nu(E)$ for every Borel set $E\subset\Omega$.

The question attributed to Kołodziej asks whether the existence of a continuous psh subsolution with zero boundary values guarantees a continuous psh solution with the same boundary values. Interpreting the stated continuity and boundary values as continuous extension to $\overline\Omega$, the proposed implication is: for every nonnegative Borel measure $\mu$ on $\Omega$, \[ \begin{gathered} \exists u\in\operatorname{PSH}(\Omega)\cap C(\overline\Omega):\qquad u:= (dd^c u)^n\geq\mu,\qquad u|_{\partial\Omega}=0 \end{gathered} \quad\Longrightarrow\quad \begin{gathered} \exists\varphi\in\operatorname{PSH}(\Omega)\cap C(\overline\Omega):\qquad (dd^c\varphi)^n=\mu,\qquad\varphi|_{\partial\Omega}=0. \end{gathered} \] The measure inequality and equality are required throughout $\Omega$.

The supplied record does not specify the intended class of domains, including boundedness, boundary regularity, or pseudoconvexity assumptions. It also does not explicitly specify the measure class, normalization of $d^c$, or boundary-trace convention. The displayed formulation makes the usual nonnegative-measure and continuous-extension interpretations explicit; it does not resolve these missing source qualifications or assert the implication for every domain in ${\mathbb{C}}^n$.

Short English statement

The question attributed to Kołodziej asks whether a continuous plurisubharmonic subsolution whose Monge--Ampère measure dominates $\mu$, with zero boundary values, ensures a continuous plurisubharmonic solution with Monge--Ampère measure exactly $\mu$ and the same boundary values. The intended domain hypotheses and precise boundary-continuity convention are missing from the supplied record.

Sources

• Local UnsolvedMath registry, permanent ID $800882$, source code catalogue entry, title: Open Problems in Pluripotential Theory --- Question 13. The supplied registry labels the problem open; this status has not been independently verified.
• Dinew, Guedj, and Zeriahi, Open Problems in Pluripotential Theory (2015), Question 13, attributed in the supplied statement to Kołodziej. Supplied source URL: https://arxiv.org/abs/1511.00705.
• Supplied provenance records access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details, not a claim of independent source consultation or verification.