Vectorial Allen–Cahn Limits Without Energy Convergence

Definitions and mathematical statement

The supplied record asks for a sharp-interface convergence result for the vectorial Allen--Cahn equation. For purposes of describing the requested target, let $u_\varepsilon$ denote the diffuse-interface solutions and let $\mathcal E$ denote the limiting multiphase configuration, where $\varepsilon\downarrow0$ denotes the interface-thickness limit. These symbols describe the intended limiting process; the record does not specify an equation, admissible solution class, or precise convergence relation.

Requested statement, subject to unresolved specifications. Establish convergence of the vectorial Allen--Cahn solutions to a De Giorgi solution of multiphase mean-curvature flow, without assuming convergence of the diffuse-interface energies to the corresponding limiting interfacial energy.

Here "unconditional" is understood only in the sense explicitly supplied: energy convergence is not an assumption. It does not remove other necessary hypotheses. The record does not specify the spatial domain or dimension, boundary conditions, time interval, number of phases, multiwell potential, equation normalization, initial-data assumptions, or initial energy bounds. It also does not define the intended De Giorgi solution, convergence topology, temporal meaning of energy convergence, or whether convergence is asserted for a subsequence or the entire family. Consequently, the supplied evidence determines this convergence target but does not determine a complete quantified mathematical conjecture.

Short English statement

The problem asks whether vectorial Allen--Cahn solutions converge to a De Giorgi solution of multiphase mean-curvature flow without an energy-convergence assumption. The supplied record leaves the hypotheses and precise notions of solution and convergence unspecified, so this is a statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005594$, source problem code catalogue entry, title: Vectorial Allen--Cahn Limits Without Energy Convergence. The record belongs to the Oberwolfach Reports Open Problems collection and lists proposed year 2023; no individual proposer is supplied.
• Supplied source citation: Partial Differential Equations (2024), Oberwolfach Reports, DOI 10.4171/owr/2023/32, pp. 1789--1842. Source URL and supplied literature-source link, labelled "Oberwolfach Report 32": https://doi.org/10.4171/owr/2023/32. The background describes these reports as documenting mathematical workshops and their open-problem discussions.
• The supplied original passage reads: "Unconditional convergence, however, remains a challenging open problem." The record says that the passage is retained verbatim as context for the extracted problem, but supplies no more specific page or contribution attribution.
• Literature assessment dated 2026-08-22: the supplied curation labels the problem open, says that the report explicitly identifies unconditional convergence as open, and states that later special-purpose numerical convergence results do not establish the general statement represented by this record. This is supplied dated curation, not an independently verified assessment; time-sensitive status and cited claims require independent verification. The only listed literature source is Oberwolfach Report 32 at the DOI link above.
• The registry labels the statement corrected_verified and says that it was reconstructed and checked against the official report. These are supplied provenance claims, not verification performed for this draft.