Uniform gradient regularity for infinity-harmonic functions

Source and verification

This article explains family 377 of the October 6, 2026 OpenAI collection. Research conclusions are attributed to its manuscripts. The review covers statements, introductions, and selected proof mechanisms; it is not an independent verification of the complete proofs.

Overview

The infinity Laplacian controls second derivatives only in the direction of the gradient. Family 377 claims a uniform interior Hölder bound for the full gradient in every dimension at least three. The exponent is positive but not explicit, and the classical model example prevents any exponent above one third.

🏷️ Preliminaries: the infinity Laplacian

For a smooth real function, define

$$\begin{array}{c}{\Delta }_{\infty }u\\ =⟨D^{2}u\nabla u,\nabla u⟩\\ =\sum _{i,j}{u}_i{u}_j{u}_{ij}.\end{array}$$

Unlike the ordinary Laplacian, this operator does not average curvature in every direction. At a point where the gradient vanishes, it degenerates completely.

Solutions are interpreted in the viscosity sense: a $C^2$ function touching $u$ from above must have ${\Delta }_{\infty }\varphi \ge 0$ at contact, and a lower touching function must have ${\Delta }_{\infty }\varphi \le 0$. This definition allows continuous functions without assuming classical second derivatives.

The equation is associated with absolutely minimizing Lipschitz extensions: on each interior subdomain, the largest local slope cannot be reduced by changing the function while preserving its boundary values. Differentiability at each point is weaker than continuity of the derivative, and a uniform Hölder estimate is stronger still.

🏷️ The manuscript’s quantitative claim

Family 377, as stated in the manuscript

For every $d\ge 3$, there are ${\alpha }_d\in (0,1/3]$ and $C_d<\infty$ such that every bounded infinity-harmonic $u\in C(B_1)$ belongs to $C^{1,{\alpha }_d}(B_{1/2})$, with

$$\begin{array}{c}\Vert \nabla u{\Vert }_{L^{\infty }(B_{1/2})}+[\nabla u{]}_{C^{0,{\alpha }_d}(B_{1/2})}\\ \le C_d{\mathrm{osc}}_{B_1}u.\end{array}$$

The exponent depends only on dimension and is obtained nonconstructively by compactness. (OpenAI, 2026)

Here $[v{]}_{C^{0,\alpha }}={\sup }_{x\ne y}|v(x)-v(y)|/|x-y{|}^{\alpha }$. The theorem does not identify ${\alpha }_d$ with the endpoint $1/3$.

🏷️ Scaling and an obstruction to better exponents

Applying the unit-ball estimate to $v(y)=u(x_0+2ry)$ and using the chain rule gives, after adjusting the constant,

$$\begin{array}{rl}& \Vert \nabla u{\Vert }_{L^{\infty }(B_r(x_0))}+r^{{\alpha }_d}[\nabla u{]}_{C^{0,{\alpha }_d}(B_r(x_0))}\\ & \le \frac{C_d}{r}{\mathrm{osc}}_{B_{2r}(x_0)}u.\end{array}$$

This records the correct units: oscillation divided by distance controls a slope.

The standard example is $u(x)=|x_1{|}^{4/3}-|x_2{|}^{4/3}$. Away from the coordinate axes, the Hessian is diagonal. Each nonzero product $u_i^2{u}_{ii}$ has magnitude $64/81$, with opposite signs for the two coordinates, so ${\Delta }_{\infty }u=0$ there. Its viscosity extension across the axes is the infinity-harmonic example cited by the manuscript; the classical calculation alone is not a check at those axes.

Along the first coordinate axis,

$$|\nabla u(h{e}_1)-\nabla u(0)|=\frac{4}{3}|h{|}^{1/3}.$$

Dividing by $|h{|}^{\alpha }$ diverges as $h\to 0$ for every $\alpha >1/3$. Adding unused coordinates gives the same obstruction in higher dimensions.

🏷️ Affine approximation at a uniform rate

The manuscript first normalizes oscillation and adjoins a linear coordinate: $U(y,b)=u(y)+b$. This preserves the infinity-harmonic equation while giving a nonzero gradient component. It avoids losing the preferred direction when the original gradient is small.

Pointwise affine blow-ups do not automatically supply one approximation rate valid for all normalized solutions and interior centers. A three-point estimate controls midpoint defects at much smaller separations, and the source uses it to obtain that uniformity.

Finer approximation is measured on thin cylinders, with axial radius $r$, transverse radius $\sqrt{\lambda }r$, and affine error $\lambda r$. A power decay of the best fitting parameter makes consecutive slopes converge at a power rate, yielding a Hölder gradient.

🏷️ The proposed Liouville contradiction

If every such power bound fails, the proof selects scales where a fit fails at one center but controlled fits succeed at surrounding centers and comparable scales. Stretching the thin directions and magnifying the error gives a nonaffine entire limit satisfying

$$D^{2}v[({\nabla }_{x}v,1),({\nabla }_{x}v,1)]=0.$$

The limit inherits uniform affine-fitting bounds on sheared boxes at every center and scale. The central Liouville claim says that every entire solution with these inherited bounds must be affine. Its proof uses quadratic suprema and infima, an averaging defect, and localization of contact points.

An affine limit would improve the deliberately failed fit, producing the contradiction. The new work lies in the uniform fitting estimates and the Liouville theorem with the exact inherited hypotheses. Compactness without those estimates would give neither a common Hölder exponent nor the stated uniform norm bound.

🏷️ Key steps and proof outlines

A three-point test controls scales smaller than the known flatness scale

Uniform closeness to an affine function on a ball does not by itself control the relative error at much smaller separations. The source addresses precisely this issue. For uniformly Lipschitz infinity-harmonic $U_i$ approaching affine functions ${\ell }_i$, it claims

$$\underset{x\ne y\in B_{1/4}}{\sup }\frac{|U_i(x)+U_i(y)-2U_i((x+y)/2)|}{|x-y|}⟶0.$$

Proof outline. If a fixed positive midpoint defect persisted, maximize a three-point expression with independently moving $x,y,z$. Write $r=|x-y|$, $c=(x+y)/2$, and $h=z-c$. The penalty is

$$B(c)\left(r-\frac{1}{8}{r}^{3/2}\right)+\frac{|h{|}^2}{r^2},$$

where $B$ localizes the maximum away from the boundary. Comparing the maximizing third point with $z=c$ gives $|h|=O(r^2)$; uniform affine closeness forces $r\to 0$.

The viscosity theorem of sums supplies three compatible second-order jets. Evaluating their matrix inequality in the directions of the three test gradients makes the infinity-Laplacian inequalities imply a nonnegative quadratic form. But the concave term has second derivative $-3/(32\sqrt{r})$; along the selected directions its contribution tends to $-\infty$, while the midpoint-penalty and localization errors remain $O(1)$. This contradiction is the asserted scale-independent midpoint control.

It lets the proof first obtain an affine blow-up at a slowly shrinking scale, then transfer its midpoint information to a much smaller offending scale. The limit there satisfies the midpoint equality and is affine too. This produces qualitative flatness uniformly over the normalized family and over moving centers. (OpenAI, 2026)

The anisotropic limit retains a nonlinear equation

The linear lift $U(y,b)=u(y)+b$ is infinity-harmonic and has an exact unit increment in the added direction. Consequently its fitted slopes remain bounded away from zero. Suppose the uniform flatness modulus has no power decay. The source selects bad scales with nearby larger scales admitting controlled cylinder fits.

A cylinder has axial radius $r$ and transverse radius $\sqrt{\lambda }r$, with error $\lambda r$. Rotate its fitted slope to the final coordinate and write

$$U(z+r(\sqrt{\lambda }x,t))=U(z)+rc(t+\lambda v(x,t)),$$

where $c$ is the slope magnitude. Substitution into the viscosity test equation gives

$$D^{2}v[({\nabla }_{x}v,1+\lambda v_t)]=0.$$

As $\lambda \to 0$, this becomes $D^{2}v[({\nabla }_{x}v,1)]=0$. Comparisons of overlapping fitted cylinders control direction changes by $O(\sqrt{\lambda })$ but slope-magnitude changes by $O(\lambda )$. Both rates are needed for compactness after this rescaling.

The resulting entire limit has affine fits with error $O(R)$ on boxes sheared by the fitted spatial slope. It must be nonaffine: an affine limit would pull back to an admissible fit at the selected bad scale, contradicting its definition. Thus power decay reduces to a Liouville theorem for this specific entire class, not to a general assertion about arbitrary solutions of the limiting equation. (OpenAI, 2026)

The final rigidity step is an exact variance calculation

The source’s intervening envelope and contact-propagation arguments produce a final limit whose forward and backward quadratic-envelope slopes equal one constant $B$ at every length. Those lengthy arguments supply the contact localization and finite limiting slope; they are inputs to the following short rigidity step.

Envelope monotonicity and growth control give the directed comparison

$$V(y,t+\ell )-V(x,t)\le B\ell +\frac{|y-x{|}^2}{2\ell }.$$

At an origin contact, let the forward and backward equality points have velocities $q_{+}$ and $q_{-}$. Comparing them directly yields

$$0\ge \frac{\ell |q_{+}{|}^2+r|q_{-}{|}^2}{2}-\frac{|\ell q_{+}+r{q}_{-}{|}^2}{2(\ell +r)}=\frac{\ell r}{2(\ell +r)}|q_{+}-q_{-}{|}^2.$$

Hence all equality velocities coincide. Comparing an arbitrary point with increasingly distant points on this common equality line forces $V(x,t)=q\cdot x+(B-|q{|}^2/2)t$. However the propagated contacts retain a strictly positive barycentric-envelope gap, impossible for an affine function. This closes the claimed Liouville contradiction. (OpenAI, 2026)

Recovering the Hölder exponent from cylinder geometry

Once $\lambda =A{r}^{\gamma }$ is available, the cylinder contains a ball of radius $\rho =\sqrt{A}{r}^{1+\gamma /2}$. Comparing successive affine fits on these inscribed balls gives slope increments $O(r^{\gamma /2})$, a summable geometric series. Thus the limiting slope exists and the Taylor remainder is $O(|h{|}^{1+\alpha })$, with

$$\alpha =\frac{\gamma }{2+\gamma }.$$

Comparing two Taylor expansions at a shared nearby evaluation point gives the same Hölder exponent for the gradients. The loss from $\gamma$ to $\alpha$ records the transverse cylinder width; it is not an arbitrary weakening of the flatness estimate.

See Also

• Family 366 uses model blow-ups to explain a sharp gradient-integrability threshold.
• Family 375 is another classification of entire limits used to recover local or global regularity.

🏷️ Manuscripts and verification scope

• Uniform Interior $C^{1,\alpha }$ Estimates for Infinity-Harmonic Functions — pinned manuscript and companion source.

No family-specific scope page was available at lean/docs/377.md in this snapshot. This records the catalogue lookup, not a claim that no related Lean code exists. No Lean build or axiom audit was performed.

OpenAI collection provides prerequisites and the reading sequence; coverage register records completed and pending families.

🐻  OpenAI 2026. Uniform Interior C^1,α Estimates for Infinity-Harmonic Functions.