Monge–Ampère Singularities for Nonpolyhedral Obstacles

Definitions and mathematical statement

For an obstacle represented by a function $\psi$, its graph is $\operatorname{graph}(\psi)=\{(x,\psi(x)):x\in\operatorname{dom}\psi\}$. A convex polytope is the convex hull of finitely many points in a finite-dimensional real affine space. These definitions explain the terminology without specifying an obstacle class absent from the record.

Research problem. Investigate singularity questions for Monge--Ampère obstacle problems when the graphs of the obstacles are not necessarily convex polytopes. The supplied original formulation asks to investigate these singularity questions; the supplied clean formulation asks to develop singular solutions in this setting. The intended scope therefore includes obstacles beyond those whose graphs are convex polytopes, but the record does not specify a precise existence, construction, or singularity-classification assertion.

Unresolved formulation. The record supplies no dimension, domain, equation or right-hand side, boundary conditions, obstacle regularity, direction of the obstacle constraint, admissible solution class, or definition of singularity. In particular, it does not specify whether solutions are understood classically, in the Alexandrov sense, or in another sense. Consequently, no fully quantified mathematical statement can be recovered without additional source information. No assertion of existence for every obstacle with a nonpolyhedral graph is made.

Short English statement

Study singularities of Monge--Ampère obstacle problems beyond the convex-polytope graph setting. The supplied clean statement emphasizes developing singular solutions, while the original wording asks more broadly to investigate singularity questions. The equation, hypotheses, and intended conclusion remain unspecified, so this is a research-program statement requiring mathematical review.

Sources

• Local UnsolvedMath registry, supplied complete record: permanent ID $30004758$, exact source problem code catalogue entry, title Monge--Ampère Singularities for Nonpolyhedral Obstacles, collection Oberwolfach Reports Open Problems. No individual proposer is supplied; the recorded proposed year is 2021.
• Supplied report citation: Partial Differential Equations (2022), Oberwolfach Reports, DOI 10.4171/owr/2021/35, pp. 1859--1914. https://doi.org/10.4171/owr/2021/35. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No item-level page or named contributor is supplied.
• Supplied literature source: Mooney--Rakshit, https://arxiv.org/abs/2204.11365. The record supplies no title or precise result identifying its relationship to the intended obstacle problem.
• Supplied literature source: Oberwolfach report, https://ems.press/content/serial-article-files/46911.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. The supplied assessment states that subsequent theory treats broad polyhedral and dual-obstacle settings, while a comparable theory for obstacles whose graphs are not convex polytopes remains unavailable. The background explicitly calls this dated literature triage derived from the supplied OWR curation and requests independent verification of time-sensitive status and cited claims.
• The registry labels the problem open and the statement corrected_verified. Its supplied verification note says that the official report was checked directly and that the clean statement preserves the complete mathematical target while restoring definitions and removing extraction artifacts or source-dependent wording. These are source-record assertions; this draft does not independently verify them or claim to have consulted the linked sources.