1 · The essentials
This work identifies seven as the first dimension in which a particular kind of energy-minimizing boundary can fail to be flat. The problem concerns a boundary whose location is itself part of what must be determined. The central result studies nonzero solutions that minimize energy everywhere and reproduce their shape when enlarged in a specific way. In dimensions up to six, all such solutions are flat; in dimension seven, a nonflat example exists. Consequently, the interior boundaries of local energy minimizers are smooth in dimensions up to six. In higher dimensions, the result limits the size of the set where smoothness fails, and in dimension seven there are only finitely many such points locally. The main result has a Lean formalization, although the supplied source does not specify its exact coverage.
A picture to help (an imperfect image, not to be taken literally)
As an imperfect image, imagine enlarging a paper template and finding that its boundary always retains the same shape. The result says that, for the particular energy-minimizing objects studied here, this kind of scaling allows only flat boundaries through dimension six, but permits a nonflat one in dimension seven. Ordinary paper cannot represent those dimensions or the energy condition.
Why it matters
The result gives an exact threshold rather than merely a range of possible dimensions. It also connects the classification of scaling-invariant global minimizers to the smoothness of boundaries for local minimizers. Above the threshold, the manuscript gives a sharp bound on the size of the singular set.
2 · Going further
The precise problem is to identify the first dimension admitting a nonflat, nonzero, one-homogeneous global minimizer of the one-phase Bernoulli energy. One-homogeneity means that for positive scale factors . The main result states that every such minimizer is flat in dimensions at most six, while a nonflat one exists in dimension seven. Thus seven is the critical dimension in this specific sense.
The abstract then states consequences for the interior free boundary of a local minimizer. It is smooth in dimensions at most six. For , its singular set has Hausdorff dimension at most , and the manuscript states that this bound is sharp. In dimension seven, the singular set is locally finite—a separate, more specific conclusion than the dimension bound alone.
The supplied abstract presents the logical progression from classification of homogeneous global minimizers to regularity and singular-set bounds for local minimizers, but it does not describe the argument establishing that classification. No companion manuscripts are supplied, so their relationship to the main result cannot be assessed here. The main result has a Lean formalization; the available information does not identify which additional consequences are included in that formalization.
Useful notions
- Energy minimization
- Global versus local minimizers
- Scaling and one-homogeneity
- Free boundaries and smoothness
- Hausdorff dimension
What was known before / what is new
Before
The source frames the question in terms of the one-phase Bernoulli energy, its minimizers, and their free boundaries. It does not state earlier dimension bounds, prior constructions, or a historical account of the critical-dimension question.
New
The central result identifies seven exactly as the first dimension admitting a nonflat, nonzero, one-homogeneous global minimizer. The manuscript also states smoothness of interior minimizing free boundaries through dimension six and a sharp singular-set dimension bound of in higher dimensions. It further states local finiteness of the singular set in dimension seven.
Limits and assumptions
The classification applies to nonzero, one-homogeneous global minimizers of the one-phase Bernoulli energy, not to arbitrary solutions or arbitrary boundaries. The regularity conclusions concern the interior free boundary of local minimizers. For , the stated dimension bound controls the size of the singular set rather than excluding singularities; dimension seven also has the stated local-finiteness conclusion. The main result has a Lean formalization, but its exact scope is not provided, so formal verification of every stated consequence should not be assumed. The supplied source does not identify remaining open problems.
Mini glossary
- Free boundary —
- A boundary whose location is determined as part of the solution rather than specified beforehand.
- One-homogeneous —
- A function is one-homogeneous when scaling its input by a positive factor scales its value by the same factor.
- Critical dimension —
- Here, the first dimension admitting a nonflat, nonzero, one-homogeneous global minimizer.
- Singular set —
- The set of free-boundary points where smoothness fails.
- Hausdorff dimension —
- A notion of dimension used to measure the size of sets, including singular sets.
- Locally finite —
- A set is locally finite when each point has a neighborhood containing only finitely many of its elements.
Keywords : One-phase Bernoulli problem · Critical dimension · Free-boundary regularity
AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.
3 · Technical details and sources
Claim made in the manuscript (source text)
Establishes seven as the first dimension admitting a nonflat, one-homogeneous global minimizer of the one-phase Bernoulli energy. Consequently, minimizing free boundaries are smooth through dimension six, and their singular sets have dimension at most in higher dimensions.
Verified by Lean: the Lean software checked the part of the reasoning described below, sometimes only part of the manuscript. This does not mean independent mathematicians have validated the result.
Manuscripts in this family
The critical dimension for one-phase Bernoulli minimizers
24/09/2026We prove that seven is the critical dimension for the one-phase Bernoulli problem: every nonzero one-homogeneous global minimizer in dimensions at most six is flat, while a nonflat one-homogeneous global minimizer exists in dimension seven. It follows that the interior free boundary of a local minimizer is smooth in dimensions at most six. In dimension n ≥ 7, its singular set has Hausdorff dimension at most , and this bound is sharp. In dimension seven, the singular set is locally finite.