Regularity of Fractional Multiphase Interfaces

Definitions and mathematical statement

A multiphase partition is a family of measurable phases that are pairwise disjoint up to null sets and together cover the prescribed ambient space up to a null set. The supplied record does not specify the ambient space, reference measure, or number of phases. Write $\sigma_{ij}$ for the surface-tension coefficient associated with phases $i$ and $j$. The usual triangle-inequality condition for such coefficients is \[ \sigma_{ij}\leq \sigma_{ik}+\sigma_{kj} \qquad\text{for distinct phase indices }i,j,k. \] This notation explains the condition mentioned in the source; the record does not specify the complete admissibility assumptions on the coefficient array.

Research problem. Develop a regularity theory for fractional multiphase minimal partitions with general surface-tension coefficients, covering both coefficients satisfying the triangle inequality and coefficients violating it, subject to the admissibility assumptions of the intended variational model.

Unresolved formulation. The supplied record does not define the fractional interaction energy, its kernel or normalization, the fractional parameter or its range, the ambient dimension or domain, the allowed phase configurations, or the meaning of minimality. It also does not specify boundary or exterior data, volume constraints, coefficient symmetry or positivity assumptions, or whether a relaxed energy is intended when the triangle inequality fails. The desired regularity conclusion, the interfaces to which it applies, and any permitted singular set are unspecified. Consequently, the record supports this research objective but does not determine a fully quantified regularity theorem. These missing specifications must be recovered from the underlying report before a precise model-specific statement can be finalized.

Short English statement

Develop a regularity theory for interfaces of fractional multiphase minimizing partitions, allowing surface tensions both with and without the triangle inequality. The supplied record leaves the energy, admissible partitions, minimality conditions, and intended regularity conclusions unspecified, so this remains a statement-only research objective requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30005993$, title Regularity of Fractional Multiphase Interfaces, exact source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. No individual proposer is named; the supplied proposed year is 2024.
• 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 attribution describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Oberwolfach Report 37/2024. https://ems.press/content/serial-article-files/50045.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open, reports that the source identifies regularity for general coefficients, including coefficients violating the triangle inequality, as open, and states that no later primary result located in that curation closes the general question. 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 that it was reconstructed and checked against the official Oberwolfach report. Those are supplied verification claims, not verification performed for this draft.