Why seven is the first dimension for singular Bernoulli minimizers

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Family 367 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Many variational problems choose both a function and the region where that function is active. The edge of that region is a free boundary: it is part of the answer rather than part of the prescribed data.

In the one-phase Bernoulli problem, energy penalizes both changes in the function and the volume of its positive region. Minimizers usually have smooth free boundaries, but in sufficiently high dimension a singular cone can also minimize energy. What is the first dimension in which this happens?

Family 367 claims that the answer is seven. More precisely, every nonzero homogeneous global minimizer in dimensions at most six is flat, while a nonflat example exists in dimension seven. The seven-dimensional example is an established result of De Silva and Jerison. The new contribution is the claimed exclusion of dimensions five and six without a symmetry assumption.

This threshold also governs singularities of general local minimizers. Below seven there would be no interior singular points; in dimension seven they would be locally finite; above seven their Hausdorff dimension would be at most the ambient dimension minus seven.

2. Background and essential definitions

An energy with an unknown active region

For a nonnegative function $v$ on a ball $B\subset\mathbb R^d$, consider

$$J(v;B)=\int_B\bigl(|\nabla v|^2+\mathbf1_{\{v>0\}}\bigr)\,dx.$$

The symbol $\mathbf1_{\{v>0\}}$ equals one in the positive region and zero elsewhere, so its integral is the volume of that region. The gradient term favors gentle variation; the volume term discourages spreading the positive phase unnecessarily.

The function belongs to $H^1(B)$, the Sobolev space of square-integrable functions with square-integrable weak first derivatives. A competitor has the same boundary values in the Sobolev trace sense. Equivalently, its difference from the original function lies in $H^1_0(B)$, the closure of compactly supported smooth functions in the $H^1$ norm.

A global minimizer on $\mathbb R^d$ minimizes this energy in every ball against all such competitors. This does not mean that its total energy over the whole space is finite. A local minimizer in an open set $D$ has the corresponding comparison property on every ball compactly contained in $D$.

Inside $\{u>0\}$, a minimizer is harmonic:

$$\Delta u=0.$$

At a smooth free boundary the normalized Bernoulli conditions are $u=0$ and $|\nabla u|=1$, with the gradient taken from the positive side. These conditions describe the balance between changing Dirichlet energy and changing positive-phase volume.

Flat solutions and cones

The simplest solution is

$$u(x)=(x\cdot e)_+, \qquad |e|=1,$$

where $s_+=\max\{s,0\}$. Its positive region is a half-space and its free boundary is a hyperplane. This is called a flat solution.

A function is one-homogeneous if

$$u(rx)=r\,u(x)\qquad(r>0).$$

Its positivity region is unchanged by dilation, hence is a cone. A nonflat homogeneous minimizer has geometry that cannot be smoothed away by simply zooming in at its vertex.

The critical dimension $d_*$ is the smallest dimension containing a nonzero, nonflat, one-homogeneous global minimizer. It concerns global energy minimality, not merely solutions of the differential equation or cones stable under a restricted set of perturbations.

Blow-ups and singular points

At a free-boundary point $x_0$, the natural rescaling is

$$u_{x_0,r}(y)=\frac{u(x_0+ry)}{r}.$$

Established compactness and monotonicity theory yields homogeneous global minimizers as limits along suitable sequences $r\downarrow0$. Such limits are called blow-ups.

A point is regular if at least one blow-up is flat. Improvement-of-flatness theory then makes the nearby free boundary smooth. A point is singular if no flat blow-up exists. Thus classifying minimizing cones controls the local geometry of far more general functions.

3. Development history and previous partial results

Alt and Caffarelli established the basic existence and regularity theory for this energy. Caffarelli’s later Harnack-inequality method connected flatness to regularity. De Silva developed another improvement-of-flatness approach, and work with Ferrari and Salsa supplied higher regularity for relevant free boundaries.

Weiss’s monotonicity and dimension-reduction theory connected singularities to homogeneous minimizing blow-ups. This explains why identifying $d_*$ is a structural question: its value determines a general bound on the size of singular sets.

Caffarelli, Jerison and Kenig obtained three-dimensional rigidity for the relevant stable homogeneous solutions. Jerison and Savin extended flatness through dimension four for stable cones with smooth spherical cross-section, implying that minimizing free boundaries are regular there. Consequently, the critical dimension was at least five.

In the other direction, De Silva and Jerison constructed a singular homogeneous global minimizer in dimension seven. Its minimality is against all Sobolev competitors with the same boundary trace, not just symmetric or homogeneous competitors. These results left

$$5\le d_*\le7.$$

Recent work by Firester, Tsiamis and Wang studies stability of cones with specified bi-orthogonal symmetry, showing a threshold at seven within that family. Their capillary-cone work constructs further minimizing examples, including another in dimension seven. Such symmetry-based classifications do not exclude arbitrary nonsymmetric minimizing cones in dimensions five and six.

Two other advances address related but distinct questions. Fernández-Real and Serra treat arbitrary entire classical stable solutions in dimension four, without a homogeneity assumption. Fernández-Real and Yu obtain regularity through dimension six for almost every parameter in an admissible increasing family of boundary data. A generic-data theorem allows exceptional parameters; the present claim concerns every minimizer in the stated class.

4. What the new manuscript claims

The main theorem states that every nonzero one-homogeneous global minimizer in $\mathbb R^d$, for $1\le d\le6$, is a half-space solution. Combining this with the established seven-dimensional example gives

$$d_*=7.$$

For every nonnegative local minimizer in an open set $D\subset\mathbb R^n$, the manuscript then claims that the interior free boundary is smooth away from its singular set $\operatorname{Sing}(u)$, with:

• No singular points when $n\le6$.
• Only finitely many singular points in every compact interior region when $n=7$.
• The bound $\dim_{\mathrm H}\operatorname{Sing}(u)\le n-7$ for $n\ge7$.

Hausdorff dimension extends the idea of dimension to irregular sets. A zero-dimensional bound alone need not mean local finiteness, so the seven-dimensional conclusion is stronger than simply substituting $n=7$ into the displayed dimension bound.

The bound is claimed sharp in every $n\ge7$. Let $U$ be the De Silva–Jerison minimizer on $\mathbb R^7$, and define

$$U_n(x,y)=U(x), \qquad (x,y)\in\mathbb R^7\times\mathbb R^{n-7}.$$

This remains a global minimizer. Its singular set contains $\{0\}\times\mathbb R^{n-7}$, giving dimension at least $n-7$; the upper bound makes the dimension exactly that value.

The theorem does not classify all seven-dimensional minimizing cones or assert that all singularities resemble one unique cone.

5. Why the advance matters

Importance: 5/5. If correct, the result determines the exact singularity threshold for a central free-boundary variational problem and yields optimal regularity conclusions in every ambient dimension.

Its force comes from the passage between scales. A statement about scale-invariant global objects rules out local singularities for arbitrary minimizers in dimensions up to six. Above the threshold, the same classification fixes the maximal dimension of the singular set.

The result also clarifies the analogy with minimal-surface theory, where curvature, stability and dimension determine when singular cones become possible. The Bernoulli problem has its own energy and boundary conditions, so that analogy motivates methods without making the two theories identical.

6. The main idea

Reduce the unknown cases to one spherical problem

Assume the first nonflat minimizing cone occurs in dimension five or six. Any singularity away from its vertex would, after another blow-up, give a lower-dimensional nonflat minimizer. By the choice of the first dimension, this cannot happen. Its intersection with the unit sphere is therefore smooth away from the special points introduced below.

A five-dimensional minimizer can be extended to six dimensions by making it independent of an extra coordinate. This preserves minimality. Its spherical cross-section acquires two suspension tips, corresponding to the two directions of the added axis. Both possible dimensions can then be studied on a region $\Omega\subset S^5$, with smooth boundary except possibly at those two tips.

Write $g$ for the restriction of the cone function to the sphere. The tensor

$$T=\nabla^2_{S^5}g+gI, \qquad w=|T|,$$

encodes the relevant angular Hessian; $I$ is the identity on tangent directions. For a flat linear solution this tensor vanishes. A nonflat component supplies a nonzero tensor somewhere.

Keep the shape of the Hessian

Stability is a necessary consequence of minimality: the second variation cannot be negative. It compares the energy of a test function with a curvature contribution on the free boundary.

Earlier approaches used a power of the Hessian norm, roughly $w^a$, as the test function. The difficulty is that interior and boundary estimates impose competing restrictions. Taking only the norm discards how the Hessian’s eigenvalues are arranged, even though that arrangement affects both estimates.

The manuscript builds two objects together: a scalar test $h$, and a vector field $Z$ whose divergence controls the test’s energy. Both depend on the normalized tensor $T/w$, together with $g$, its gradient and suitable derivative data. Keeping the same tensor information in both constructions is the proposed way around the incompatible estimates.

Create a strict surplus against stability

With a fixed exponent $a\in(1/2,1)$, the claimed pointwise inequalities have the form

$$\operatorname{div}Z \ge |\nabla h|^2+4h^2+\varepsilon_*w^{2a},$$

and, on the regular boundary,

$$N\cdot Z+Hh^2\ge0.$$

Here $\varepsilon_*>0$, $N$ is the inward unit normal, and $H$ is the specified sum of principal curvatures. The positive term $\varepsilon_*w^{2a}$ is the decisive surplus. After integration, the divergence theorem and stability give incompatible bounds unless the Hessian vanishes.

The coefficient four comes from the six-dimensional spherical stability formulation. These are dimension-specific estimates, not inequalities asserted on arbitrary manifolds.

Two cancellations reduce the calculation to finitely many normalized tensors. Interior contractions cancel fourth derivatives; boundary reflection removes derivative components not determined by the boundary conditions. The manuscript supplies weighted sums of tensor squares and rational remainder bounds for the resulting algebraic inequalities. These finite calculations are part of its stated argument.

Do not ignore the places where normalization fails

The expression $T/w$ is undefined where $w=0$. The suspension tips also prevent direct integration over a wholly smooth compact region. Removing these sets without control would leave a gap in the intended comparison.

The manuscript uses a regularized Bochner identity to obtain an integrable derivative weight near Hessian zeros. Explicit suspension formulas provide integrable powers of distance near the two tips. Smooth cutoffs then remove both kinds of exceptional set, with errors tending to zero.

A genuinely nonflat component retains positive surplus in the limit, contradicting stability. This excludes both candidate dimensions. Standard dimension reduction and the known seven-dimensional example then yield the sharp singular-set statement.

7. Scope, limitations, and open questions

The energy has constant volume coefficient one and a single nonnegative phase. Variable coefficients, two-phase problems and other free-boundary energies require their own statements. The conclusions are interior; boundary-contact behavior is not included.

The central classification concerns homogeneous global minimizers. It does not say that every entire stable solution in dimensions five or six is one-dimensional, nor that every stationary cone is flat. Minimality against arbitrary Sobolev competitors is the operative hypothesis.

In dimension seven, local finiteness allows infinitely many singular points approaching the boundary of the ambient domain or escaping to infinity. In higher dimensions, the Hausdorff bound alone is not a complete structural description of the singular set.

Classification of all cones at and above the threshold, uniqueness of singular blow-ups and sharper quantitative geometry remain distinct questions from identifying the critical dimension.

8. Sources and further reading

The primary papers linked above trace the Alt–Caffarelli theory, Weiss’s dimension reduction, Jerison–Savin rigidity and the De Silva–Jerison singular example. Velichkov’s Regularity of the One-phase Free Boundaries develops the background used to pass from cone classification to local regularity.

The manuscript below supplies the claimed exclusion in dimensions five and six. Its finite tensor calculations are reported as part of the source’s argument, not independently certified here.

Source manuscripts

• M1. OpenAI, The critical dimension for one-phase Bernoulli minimizers (September 24, 2026).

Sources consulted on 26 October 2026 at repository revision 1581d4c. Download the manuscript citations.