Rigidity, Regularity, and Vectorial Nucleation in Nonlinear Inclusions

Definitions and mathematical statement

The supplied record concerns geometrically nonlinear differential inclusions, meaning constraints that require derivatives of an unknown map to belong to prescribed sets. It does not specify the unknown map, domain, derivative order, constraint sets, admissible function spaces, or boundary conditions. Consequently, the following is a qualified research agenda rather than a fully quantified conjecture.

1. For geometrically nonlinear differential inclusions, determine whether there is a critical threshold separating rigid and flexible regimes. The source does not identify the parameter being varied, define either regime, specify a class of inclusions over which the question is quantified, or describe behavior at the proposed threshold.
2. Determine whether one can bound the maximal regularity of irregular solutions. The source does not define irregularity, specify a regularity scale, identify the solutions under consideration, or state the dependence or intended sharpness of such a bound.
3. Determine whether scalar nucleation models for shape-memory alloys can be extended to vectorial models incorporating compatibility conditions, and clarify their relation to convex integration. The scalar models, vectorial unknowns, compatibility conditions, and required meaning of extension are not supplied.
4. Determine whether the resulting models can be analyzed through associated random processes. Neither the stochastic construction nor the quantities or properties to be analyzed are specified.

These questions retain the scope of the supplied passage. A precise mathematical statement requires additional model specifications, hypotheses, boundary conditions where applicable, and quantifiers; none can be selected uniquely from this record.

Short English statement

The source asks about rigidity--flexibility thresholds, limits on the regularity of irregular solutions, and vectorial extensions of scalar shape-memory-alloy nucleation models, including compatibility, convex integration, and stochastic descriptions. It supplies a broad research agenda without enough model details for a fully quantified conjecture.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30004481$, exact source problem code catalogue entry, in the collection Oberwolfach Reports Open Problems; title: Rigidity, Regularity, and Vectorial Nucleation in Nonlinear Inclusions. No individual proposer is identified; the supplied proposed year is 2020.
• Supplied report citation: Calculus of Variations (2021), Oberwolfach Reports, pp. 1139--1196, DOI 10.4171/owr/2020/22, https://doi.org/10.4171/owr/2020/22. The background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No specific passage, contribution, or contributor is identified for these questions.
• Supplied literature source: OWR Calculus of Variations report, https://ems.press/content/serial-article-files/46860.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open and reports finding no general resolution of the broad questions concerning rigidity--flexibility thresholds, regularity, vectorial nucleation, and stochastic descriptions. This is supplied dated curation, not an independently verified literature assessment. Its time-sensitive status and cited claims require independent verification.
• The registry labels the statement corrected_verified and reports a manual semantic, typographic, and LaTeX review asserting self-containedness and preservation of the target. This is supplied review metadata; it does not resolve the missing mathematical specifications listed here.