Regularity of Stationary Constraint Maps and Their Heat Flow

Definitions and mathematical statement

Let $\mathcal A$ denote an admissible class of maps from a source domain into a constrained target, and let $E$ denote their energy. These are placeholders: the supplied record does not specify the source domain, target geometry, whether the target boundary is included, function space, energy functional, or boundary conditions. An energy-minimizing map $u\in\mathcal A$ satisfies \[ E(u)\leq E(v)\qquad\text{for every }v\in\mathcal A. \] A stationarity condition must specify the permitted variations. For a chosen family of differentiable two-sided admissible variations $t\mapsto u_t$, with $u_0=u$, its schematic form is \[ \left.\frac{\mathrm d}{\mathrm dt}E(u_t)\right|_{t=0}=0 \qquad\text{for every permitted variation.} \] This schematic condition does not resolve the source's meaning of stationary constraint map: the record does not identify the variations or explain how they interact with the target constraint. In particular, a one-sided admissible variation requires a separate variational formulation.

Requested research problem. Determine which regularity conclusions available for energy-minimizing constraint maps extend to stationary constraint maps, under the intended hypotheses and notion of stationarity, and develop the corresponding regularity theory for the associated parabolic heat flow.

Unresolved qualifications. The supplied passage refers to "the same regularity issue" without supplying the preceding regularity result or its hypotheses. It also leaves the heat-flow equation, solution concept, initial and boundary data, and intended regularity conclusions unspecified. Consequently, the record supports this research direction but does not determine a unique quantified regularity theorem or a definite parabolic initial-boundary value problem. The placeholders above introduce no additional hypotheses or asserted regularity conclusions.

Short English statement

The problem asks how regularity results for energy-minimizing maps subject to a target constraint extend to stationary maps and to their associated heat flow. The supplied excerpt omits the precise variational setting and regularity targets, so a complete theorem statement requires further source context. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $30005992$, title: Regularity of Stationary Constraint Maps and Their Heat Flow; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and attributes the problem to workshop open-problem discussions. It lists proposed year $2024$ and supplies no individual proposer.
• Supplied source citation: Calculus of Variations (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/37, pp. 2113--2178. https://doi.org/10.4171/owr/2024/37. The supplied passage asks about regularity beyond energy-minimality, specifically for stationary constraint maps, and about the corresponding heat-flow problem. Its reference to an earlier regularity issue is unresolved in the supplied record.
• Supplied literature source: Constraint Maps, 2026. https://link.springer.com/article/10.1007/s44007-026-00209-w.
• Supplied literature source: Oberwolfach Report 37/2024. https://ems.press/content/serial-article-files/50045.
• Literature assessment date: 2026-08-22. This is supplied dated curation, not independently verified. The supplied assessment labels the problem open and states that the extension to stationary maps and the corresponding heat-flow theory remain unresolved and are still presented as open directions in current surveys. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that it was reconstructed and checked against the official Oberwolfach report. These are supplied provenance claims, not independent verification by this draft. The citation's $2025$ designation and the report's $2024$ identifiers are retained as supplied.