Open Problems in Pluripotential Theory — Question 43

Definitions and mathematical statement

Write $Q_T=[0,T]\times X$ and $\Gamma_0=\{0\}\times X$, the set called the parabolic boundary in the supplied statement. The space $X$, the terminal time $T$, and the degenerate complex Monge--Ampère flow are contextual data whose definitions and hypotheses are absent from this record. In particular, the phrase "the above degenerate complex Monge--Ampère flow" refers to an equation not included in the supplied extract.

Comparison question attributed to Eyssidieux--Guedj--Zeriahi. For the space, time interval, and flow intended in the source, is it true that every viscosity subsolution $u$ and every viscosity supersolution $v$ of that same flow satisfy \[ \left[u(0,x)\leq v(0,x)\text{ for every }x\in X\right] \quad\Longrightarrow\quad \left[u(t,x)\leq v(t,x)\text{ for every }(t,x)\in[0,T]\times X\right]? \] The intended domains and admissible function classes for $u,v$, the viscosity definitions, and any conditions at the terminal time must be supplied by the missing context. The conclusion includes $t=T$, as in the supplied statement. This is a contextual comparison question, not a fully specified standalone PDE problem; no missing flow equation or hypotheses are being imposed here.

Short English statement

The question asks whether initial pointwise ordering between a viscosity subsolution and a viscosity supersolution of the intended degenerate complex Monge--Ampère flow persists throughout $[0,T]\times X$. The flow and its hypotheses are missing from the supplied extract, so a complete standalone formulation requires the surrounding source context.

Sources

• Local UnsolvedMath registry, permanent ID $800911$, exact source problem code catalogue entry; title: Open Problems in Pluripotential Theory --- Question 43. The registry records Dinew, Guedj, and Zeriahi as proposers and gives the year 2015; the extracted question specifically attributes the comparison question to Eyssidieux--Guedj--Zeriahi.
• Supplied source reference: Dinew, Guedj, Zeriahi, Open Problems in Pluripotential Theory (2015), Question 43, https://arxiv.org/abs/1511.00705. The record reports extraction by arxiv_latex and an access date of 2026-10-16. This access information is supplied provenance and was not independently verified here.
• The registry labels the problem open. This is a supplied status label, not an independently verified assessment of the literature or current mathematical status.