Angle Dependence of One-Phase Bernoulli Singularities

Definitions and mathematical statement

The supplied record asks whether the sharp dimension bound for singularities can depend on the contact angle. It juxtaposes the one-phase Bernoulli problem with boundary regularity for capillary minimizers, but does not specify a mathematical identification between these settings.

A conditional formulation of the intended question is as follows. Let $n$ denote the surface dimension, let $\Theta$ be the admissible set of contact angles, and, once the variational problem and boundary regularity notion have been specified, let $\mathcal C_{n,\theta}$ be its class of admissible minimizing configurations at angle $\theta$. Write $S_{\partial}(E,\theta)$ for the singular set of the free surface at the container boundary. Define the optimal universal boundary singular-set dimension by \[ D_n(\theta):=\sup_{E\in\mathcal C_{n,\theta}}\dim_{\mathrm H}S_{\partial}(E,\theta), \] where $\dim_{\mathrm H}$ denotes Hausdorff dimension; for this conditional definition, set $\dim_{\mathrm H}\varnothing=-\infty$ and $\sup\varnothing=-\infty$. The angle-dependence question is whether \[ \exists n\;\exists\theta_1,\theta_2\in\Theta \quad D_n(\theta_1)\ne D_n(\theta_2). \] This is a proposed precise reading, conditional on specifying $\Theta$, $\mathcal C_{n,\theta}$, and $S_{\partial}$. Those objects cannot be recovered uniquely from the supplied excerpt. In particular, this formulation must not be treated as an established equivalence between a Bernoulli problem and a capillary problem.

For context, the supplied excerpt describes a container $\Omega$ of dimension $n+1$, a set $E\subset\Omega$ minimizing an unspecified functional $G$, possibly subject to a volume constraint, and the free surface \[ M:=\partial E\cap\operatorname{int}\Omega. \] It states that there is $\varepsilon(n)>0$ such that $M$ is smooth away from a singular set and gives the following estimates, reproduced in the source's notation: \[ \begin{aligned} \dim(\operatorname{sing}M\cap\operatorname{int}\Omega)&\le n-7,\\ \dim(\operatorname{sing}M\cap\partial\Omega)&\le n-4,\\ \dim(\operatorname{sing}M\cap\partial\Omega)&\le n-7 &&\text{if }|\sigma|<\varepsilon(n),\\ \dim(\operatorname{sing}M\cap\partial\Omega)&\le n-5 &&\text{if }|\sigma-1|<\varepsilon(n) \text{ or }|\sigma+1|<\varepsilon(n). \end{aligned} \] The excerpt also attributes an earlier boundary estimate $n-3$ to its references [1] and [3], and the improvements to its reference [2]. It mentions improved estimates near contact angles $0^\circ$, $90^\circ$, and $180^\circ$, but supplies neither a definition of $\sigma$ nor its relation to the angle. Since $M$ as defined lies inside $\Omega$, the displayed boundary singular-set notation requires an intended extension to the closure of the free surface; that extension and its regularity criterion are not defined in the record. These estimates are supplied contextual claims, not independently verified assertions here.

The source further describes the sharp one-phase Bernoulli singular-dimension bound as belonging to $\{n-5,n-6,n-7\}$. Its dated assessment describes the corresponding critical dimension as belonging to $\{5,6,7\}$. The relation between these dimension conventions and the capillary boundary problem remains unspecified. Missing hypotheses on $G$, $\Omega$, admissible minimizers, and boundary regularity prevent an unconditional precise statement.

Short English statement

Can the sharp singular-set dimension bound vary with the contact angle? The supplied record connects this question to one-phase Bernoulli free boundaries and capillary minimizers, but does not define the variational problem or the connection precisely enough to identify a unique mathematical target. The displayed formulation is conditional and requires human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006412$, title Angle Dependence of One-Phase Bernoulli Singularities, exact source problem code catalogue entry. No individual proposer is supplied; the proposed year is 2025. The supplied status is open. The supplied statement status is corrected_verified, with a qualification that only formula delimiters, notation, grammar, and typography were repaired; this is not an independent mathematical verification.
• Supplied report citation: Differentialgeometrie im Großen (2026), Oberwolfach Reports, DOI 10.4171/owr/2025/30, pp. 1591--1646, https://doi.org/10.4171/owr/2025/30. The literature list labels the same link Oberwolfach Report 30/2025. The discrepancy between the title year 2026 and the supplied 2025 report identifiers is unresolved. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature source: Area-minimizing capillary cones, https://arxiv.org/abs/2601.18794.
• Supplied literature source: Improved regularity for minimizing capillary hypersurfaces, https://arxiv.org/abs/2401.08028.
• Literature assessment checked at 2026-08-22: the supplied curation labels the problem open, states that the one-phase Bernoulli critical dimension is known only to lie in $\{5,6,7\}$, and says that current angle-dependent capillary results do not determine whether the optimum changes with angle. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment. Time-sensitive status and cited claims require independent verification.
• The excerpt's references [1], [2], and [3] have no supplied bibliographic mapping. No individual author or theorem attribution can reliably be assigned to those labels from this record alone.