Boundary Frequency Gap for Optimal Partitions

Definitions and mathematical statement

Let $d\in{\mathbb{N}}$, $d>2$. The supplied record concerns a spectral optimal partition of a domain $D\subset{\mathbb{R}}^d$, its associated eigenfunctions $u_1,\ldots,u_N$, and a free interface $\mathcal F$. A boundary free-interface point is intended to be a point where the free interface reaches $\partial D$; the precise definition of this boundary contact set is not supplied. Write $\gamma(x_0)$ for the limiting frequency at such a point, whenever the relevant frequency function and its limit are defined. The original excerpt contains a corrupted integral formula, so neither the exact frequency function nor its normalization can be recovered reliably from this record.

For a fully specified admissible class of such partition problems in dimension $d$, let \[ \Gamma_d=\{\gamma(x_0):x_0\text{ is an admissible boundary free-interface point}\}. \] Interpreting $\delta_d$ as the first possible frequency level strictly above $2$, define conditionally \[ \delta_d:=\inf\left(\Gamma_d\cap(2,\infty)\right), \qquad \inf\varnothing:=+\infty. \] This interpretation makes the conjectural value in the supplied statement meaningful; the record does not provide an uncorrupted definition of $\delta_d$. In particular, this quantity is a frequency threshold, whereas the corresponding numerical separation from $2$ would be $\delta_d-2$.

Conjectural target, conditional on the missing admissibility and frequency definitions. For every integer $d>2$, determine $\delta_d$, and prove or disprove that \[ \Gamma_d\cap(2,5/2)=\varnothing, \qquad \frac52\in\Gamma_d, \qquad\text{hence}\qquad \delta_d=\frac52. \] The supplied conjecture further specifies that the value $5/2$ is attained by three partition domains forming a triple junction of $3/2$-points arranged along a line that reaches $\partial D$ orthogonally. The record does not define a $3/2$-point or specify whether this geometry describes the original partition, a tangent configuration, or a homogeneous blow-up. Those alternatives must be resolved before the attainment clause becomes a precise geometric assertion.

The spectral objective, admissible partition class, number of phases, boundary conditions, domain hypotheses, frequency definition, and boundary contact notion are missing. Consequently, the displayed formulation records a conditional target rather than a complete unconditional conjecture. The excerpt emphasizes technical and topological boundary difficulties: frequency monotonicity is problematic even for $C^{1,\alpha}$ boundaries; defining the optimal partition on the boundary requires analysis of eigenfunctions near $\partial D$; and regular free-boundary smoothness is described as relying on blow-up decay obtained through epiperimetric inequalities, rather than an implicit-function-theorem argument. These comments do not themselves specify the hypotheses of the conjecture.

Short English statement

For spectral optimal partitions in dimension $d>2$, determine the sharp boundary frequency threshold above $2$. The supplied conjecture predicts the threshold $5/2$, attained by a three-domain junction whose $3/2$-points lie along a line meeting the boundary orthogonally. The admissible problems, frequency normalization, and precise junction geometry remain unspecified, so this is a conditional statement-only draft requiring mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005990$, title Boundary Frequency Gap for Optimal Partitions, exact source problem code catalogue entry; collection: Oberwolfach Reports Open Problems. The record supplies proposed year $2024$, but no named proposer.
• Supplied report citation: Calculus of Variations (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/37, pp. 2113--2178. Source URL and supplied literature-source link, labeled "Oberwolfach source report": https://doi.org/10.4171/owr/2024/37. This report is supplied as primary provenance; its contents have not been independently consulted here. The citation's $2025$ label and its $2024$ DOI component are retained as supplied.
• Literature assessment dated exactly 2026-08-22: the record labels the problem open and reports that only a positive dimension-dependent boundary frequency gap is known, while the conjectural threshold $5/2$ remains unproved and is listed as open in the cited report. This is supplied dated curation, not an independently verified mathematical or literature assessment.
• The supplied registry labels the statement corrected_verified and says that it was reconstructed and checked against the official Oberwolfach report. This is a source-provided verification claim, not verification performed for this draft. The damaged excerpt refers to reference [8], but its bibliographic entry is absent.