How smooth is the least-steep extension of a function?

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Family 377 · Partial differential equations
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-26

1. The question

Suppose heights are prescribed around the edge of a region. We want to fill in the interior while making the steepest slope as small as possible. We also want the same optimality in every smaller region, with its own boundary values held fixed.

This is an absolute minimization problem. Its solutions are called infinity-harmonic functions. They need not be twice differentiable, even when considered well inside the region. The central regularity question is how smoothly their slopes vary.

A tangent plane can exist at every point without those tangent planes depending continuously on the point. Earlier work established everywhere differentiability in all dimensions. The manuscript in this family claims the stronger conclusion that the gradient is locally Hölder continuous in every dimension at least three, with a uniform estimate depending only on dimension and the function’s oscillation.

The exponent is positive but unspecified. The manuscript does not claim the sharp exponent $1/3$, nor any regularity at the boundary.

2. Background and essential definitions

Minimizing the largest slope

A function is Lipschitz if there is a finite constant $L$ such that

$$|u(x)-u(y)|\leq L|x-y|.$$

This says that changes in height never exceed $L$ times distance. For a sufficiently regular function, the local version is a bound on $|\nabla u|$, the magnitude of its gradient.

An absolute minimizer cannot improve its largest gradient by changing the function inside any relatively compact subregion $V$ while leaving its boundary values fixed. For every Lipschitz competitor $w$ with the same boundary values, it satisfies

$$\|Du\|_{L^\infty(V)}\leq\|Dw\|_{L^\infty(V)}.$$

Here $D$ denotes the gradient, and the $L^\infty$ norm is the essential supremum. “Absolute” emphasizes local optimality on every subregion, stronger than just solving one global extension problem.

The infinity Laplacian

For a twice differentiable function, the relevant equation is

$$\Delta_\infty u =\langle D^2u\,\nabla u,\nabla u\rangle =\sum_{i,j=1}^d u_i u_j u_{ij}=0.$$

The matrix $D^2u$ is the Hessian. Only its component along the gradient is tested. By comparison, the usual Laplacian sums second derivatives in all coordinate directions. The infinity Laplacian is therefore highly degenerate: it does not directly constrain all directions of bending.

Continuous solutions are interpreted in the viscosity sense. If a smooth test function $\varphi$ touches $u$ locally from above at $z$, one requires $\Delta_\infty\varphi(z)\geq0$; touching from below requires the opposite inequality. This is the sign convention for the unnormalized operator used here. Tests with zero gradient are included.

Viscosity solutions need not have classical second derivatives. The theory allows the equation to express the minimizing property even at nonsmooth points.

What Hölder continuity of the gradient means

A function is $C^{1,\alpha}$ when it has a continuous gradient satisfying a local estimate

$$|\nabla u(x)-\nabla u(y)|\leq C|x-y|^\alpha$$

for some $\alpha>0$. The exponent quantifies how quickly neighboring tangent planes approach one another.

The Hölder seminorm is

$$[g]_{C^{0,\alpha}(E)} =\sup_{x\ne y\text{ in }E}\frac{|g(x)-g(y)|}{|x-y|^\alpha}.$$

The oscillation $\operatorname{osc}_E u=\sup_Eu-\inf_Eu$ measures vertical variation. It is unchanged when a constant is added to $u$, as a gradient estimate should be.

3. Development history and previous partial results

Aronsson developed absolute minimization and its differential equation in the 1960s. Jensen later established the viscosity characterization and uniqueness for the associated Dirichlet problem. Crandall, Evans and Gariepy developed comparison with cones, an equivalent way to characterize solutions using simple functions shaped like distance cones.

These results supply existence, uniqueness and local slope control, but finer regularity is a separate problem. Crandall and Evans showed that infinitesimal rescalings at a fixed point have affine limits. An affine function has the form $a+p\cdot x$: a constant plus a plane with slope $p$. Knowing that limits are planes does not by itself supply a common rate of approximation, or identify the same slope along every sequence of scales.

In the plane, Savin proved continuous differentiability in 2005. Evans and Savin obtained interior $C^{1,\alpha}$ regularity for some positive exponent in 2008. Their method introduced thin-cylinder rescalings and a limiting equation that also appear in the present manuscript.

Evans and Smart proved everywhere differentiability in every dimension in 2011. This identifies a derivative at each point. It does not, by itself, imply that derivatives at nearby points are close. A further higher-dimensional special case is the work of Peng, Zhang and Zhou on infinity-harmonic potentials in convex rings, where they established interior continuous differentiability and nonvanishing of the gradient.

The planar endpoint has recent developments. Brustad’s June 2026 preprint gives $1/3$-Hölder continuity under additional assumptions that the gradient is nonzero and has a Lipschitz inverse. Xu’s July 2026 preprint states unrestricted planar local $C^{1,1/3}$ regularity, while separately leaving a universal endpoint norm estimate open.

The present October 4 manuscript concerns arbitrary solutions in dimensions $d\geq3$, and its main statement is uniform over a normalized family of solutions. It seeks a positive dimension-dependent exponent, not the planar endpoint conclusion.

4. What the new manuscript claims

For every integer $d\geq3$, Theorem 1.1 asserts the existence of constants

$$0<\alpha_d\leq\frac13,\qquad C_d<\infty,$$

depending only on $d$, such that every bounded continuous infinity-harmonic function on the Euclidean unit ball $B_1\subset\mathbb R^d$ belongs to $C^{1,\alpha_d}(B_{1/2})$ and obeys

$$\|\nabla u\|_{L^\infty(B_{1/2})} +[\nabla u]_{C^{0,\alpha_d}(B_{1/2})} \leq C_d\operatorname{osc}_{B_1}u.$$

No nonvanishing-gradient assumption is imposed. Neither smooth boundary data nor a special shape for distant parts of the domain enters this interior estimate.

Rescaling gives the corresponding statement in a ball $B_{2r}(x_0)$ contained in an open domain:

$$\|\nabla u\|_{L^\infty(B_r(x_0))} +r^{\alpha_d}[\nabla u]_{C^{0,\alpha_d}(B_r(x_0))} \leq \frac{C_d}{r}\operatorname{osc}_{B_{2r}(x_0)}u.$$

The powers of $r$ reflect the change in scale: gradients measure height per unit distance. In particular, every continuous infinity-harmonic function on an open subset of $\mathbb R^d$ is locally continuously differentiable. Global boundedness of the function on that entire domain is unnecessary, because the estimate can be applied on smaller balls.

The constants and exponent arise through compactness. The manuscript supplies neither a numerical value of $\alpha_d$ nor an endpoint claim that $\alpha_d=1/3$.

5. Why the advance matters

Importance: 5/5. If correct, a uniform positive Hölder exponent for arbitrary higher-dimensional solutions resolves a central regularity problem for a fundamental degenerate equation. It upgrades the existence of a tangent plane at each point to quantitative control of how those planes vary.

Uniformity is part of the advance. For all solutions with oscillation at most one on $B_1$, the same dimension-dependent constants control the gradient on $B_{1/2}$. A statement assigning an unrelated modulus of continuity to each individual solution would be weaker.

The regularity ceiling has a concrete illustration. Aronsson’s example is

$$u(x)=|x_1|^{4/3}-|x_2|^{4/3},$$

independent of any additional coordinates. It is infinity-harmonic, and along the first coordinate axis its gradient changes by

$$|Du(he_1)-Du(0)|=\frac43|h|^{1/3}.$$

Here $e_1$ is the first coordinate unit vector. Thus no exponent greater than $1/3$ can hold for all solutions, even locally. The claimed theorem gives some positive exponent below or at this natural ceiling, without determining the best one.

6. The main idea

Turn individual flatness into uniform flatness

The manuscript first normalizes the oscillation and adds one linear coordinate:

$$U(y,b)=u(y)+b.$$

The lifted function remains infinity-harmonic, and its affine approximations have a nonzero slope because of the extra coordinate. This avoids dividing the argument into regions where the original gradient vanishes and regions where it does not.

A three-point estimate controls a midpoint’s departure from the average of two endpoint values. Its relevance is that it works even when the endpoints are much closer together than the scale on which the function is already approximately affine.

Combining that control with affine blow-ups at fixed centers gives uniform small-scale approximation over all normalized solutions and all interior centers. Ordinary compactness of the functions alone would not provide such a common rate for their slopes.

Seek a power rate using thin cylinders

Next the argument uses cylinders elongated along an approximate slope. A fitting parameter $\lambda$ allows axial size $r$, transverse size $\sqrt\lambda\,r$, and error $\lambda r$. The geometry distinguishes directions that the equation treats differently.

If the fitting error decreases at a uniform power rate, successive affine slopes converge at a power rate too, producing the Hölder estimate. Suppose instead that every such bound fails. The manuscript selects nearly worst scales and centers, with enough control on nearby fits to take a limit after anisotropic rescaling.

The limiting function $v(x,t)$, defined for all $(x,t)\in\mathbb R^m\times\mathbb R$, solves

$$D^2v[(\nabla_xv,1)]=0, \qquad D^2v[\xi]=\xi^{\mathsf T}D^2v\,\xi.$$

The coordinate $t$ is an axial variable of the rescaled elliptic problem, not physical time. In addition, the limit has controlled affine fits at every center and radius, in boxes sheared according to their fitted horizontal slopes.

A Liouville theorem is a rigidity statement for an entire solution. The required theorem says that every limit with these equation and fitting properties must be affine. That contradicts the retained failure of improvement and yields a power rate. The added fit assumptions are essential: the claim is not that every entire solution of the limiting equation is affine.

Compare values through quadratic envelopes

To establish that rigidity, the manuscript uses forward and backward quadratic envelopes. For example,

$$P_\ell v(x,t) =\sup_y\left\{v(y,t+\ell)-\frac{|y-x|^2}{2\ell}\right\}, \qquad \ell>0.$$

The quadratic penalty prevents distant points from contributing without cost. A backward envelope uses an infimum at $t-\ell$, with the quadratic penalty added instead. The affine-fit bounds control growth so that these extrema are attained.

Appropriate envelope difference quotients are monotone in $\ell$ and share a small-length limit at each point. Constancy of both slopes across all lengths at one point forces affinity. The challenge is to reach such a situation while retaining evidence that the limiting function is nonaffine.

Preserve a defect while moving toward one terminal point

For an affine function, the average of its values equals its value at the average of the input points. The argument measures failure of this identity by a horizontal averaging defect. It normalizes configurations approaching the largest relevant defect ratio and obtains an extremal solution.

A variance-penalized averaging envelope then gives a subsolution touching the original solution at a positive axial coordinate, while retaining a positive gap before a linear correction. The variance penalty is chosen so that the viscosity inequality survives the averaging operation.

Forward and backward envelope optimizers define two possible steps. A discrete averaging argument controls the accumulated squared discrepancy of their velocities. This propagates contacts toward zero axial coordinate while keeping one common terminal point inside controlled fitted frames.

Using the same terminal point matters: there, both envelope slopes have one finite small-length limit. After rescaling the successive contacts, differences of slopes at fixed lengths converge to zero. The final limit therefore has constant envelope slopes and must be affine, yet it still carries the positive averaging defect. The manuscript uses this contradiction to prove its Liouville theorem and then the regularity estimate.

7. Scope, limitations, and open questions

The statement is interior regularity for the unnormalized infinity Laplace equation in Euclidean space. It does not give boundary estimates, an explicit Hölder exponent, or the sharp $1/3$ endpoint in dimensions at least three.

It also does not claim twice differentiability. Aronsson’s example already shows why a smoothness conclusion comparable to ordinary harmonic functions would be inappropriate.

Everywhere differentiability, continuous differentiability, qualitative Hölder continuity for each solution, and a uniform oscillation-controlled Hölder estimate are distinct levels of information. The main theorem claims the last of these, with constants depending only on dimension.

Determining the optimal higher-dimensional exponent and obtaining endpoint estimates remain natural questions beyond the stated result. Extensions to more general absolute-minimization equations would require their own hypotheses and arguments. The analytical claims here are explained as presented by the manuscript, without independent proof verification.

8. Sources and further reading

• Aronsson, Extension of functions satisfying Lipschitz conditions, and his singular-solution paper: the variational foundations and the regularity obstruction.
• Jensen, Uniqueness of Lipschitz extensions: minimizing the sup norm of the gradient: the viscosity and uniqueness theory.
• Savin, C1 regularity for infinity harmonic functions in two dimensions, and Evans–Savin, their planar Hölder regularity paper: earlier planar results and the anisotropic method.
• Evans and Smart, Everywhere differentiability of infinity harmonic functions: the prior all-dimensional differentiability result.
• Peng, Zhang and Zhou, Regularity from p-harmonic potentials to infinity-harmonic potentials in convex rings: a higher-dimensional special class.
• Brustad, Infinity-harmonic functions in the plane: Regularity by injectivity, and Xu, Infinity-harmonic functions and inverse mean curvature flow clusters: the recent planar endpoint work distinguished above.

Source manuscripts

• M1. OpenAI, Uniform Interior C^{1,\alpha} Estimates for Infinity-Harmonic Functions (October 4, 2026).

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