🔢 mathematics

Infinity-harmonic functions in the plane: Regularity by injectivity

This paper proves that $\infty$ -harmonic functions in the plane possess a $1/3$ -Hölder continuous gradient by leveraging a connection to the one-dimensional heat equation under a specific injectivity condition on the gradient.

Original authors: Karl K. Brustad

Published 2026-06-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Karl K. Brustad

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The "Perfectly Smooth" Problem

Imagine you are a landscape architect trying to design a surface (like a hill or a valley) that follows a very specific, strict rule: The surface must be as smooth as possible in every direction, but it cannot have any "bumps" that curve in two directions at once.

In math, this is called an $\infty$ -harmonic function. These shapes are famous because they appear in game theory (like the "Tug-of-War" game) and image processing.

For a long time, mathematicians have been arguing about how "smooth" the slope of these hills is.

We know the hills are continuous (no cliffs).
We know the slope changes smoothly most of the time.
The Big Question: Is the slope always changing in a predictable, gentle way? Specifically, does the slope change at a rate of $1/3$ power? (Think of this as a specific "smoothness limit").

There is a famous example (Aronsson's solution) that looks like a sharp corner made of four curved slices. This example proves that you can't expect the slope to be perfectly smooth everywhere. But, mathematicians suspect that $1/3$ -smoothness is the absolute best you can ever hope for.

The Paper's Main Idea: The "One-Way Street" Rule

Karl Brustad's paper doesn't try to prove this for every possible hill. Instead, he focuses on a special, slightly easier situation to crack the code.

He introduces a condition called "Metric Injectivity."

The Analogy:
Imagine the slope of your hill is a compass needle pointing in a specific direction.

Normal Situation: On a complex hill, you might walk in a circle and the compass needle spins around, pointing in every direction.
Brustad's Condition: He asks, "What if the compass needle is a one-way street?" Meaning, if you see a needle pointing North-East, there is only one specific spot on the map where that needle exists. No two spots share the exact same slope direction.

Brustad says: "If your hill has this 'one-way street' property for its slopes, then I can prove the slope changes exactly at that $1/3$ -smoothness limit."

The Magic Trick: Turning a Hill into a Heat Wave

The most clever part of the paper is how he proves this. He doesn't just look at the hill; he transforms it into something completely different: Heat.

The Transformation: He takes the geometry of the hill and stretches it out into a new coordinate system. In this new world, the complex rules of the hill turn into the Heat Equation.
- The Heat Equation describes how heat spreads out over time. It is a very well-behaved, predictable, and "calm" equation.
The Connection: By proving that the hill's slope behaves like heat spreading out, he can use all the known, powerful tools of heat physics to solve the problem.
The Result: Because heat spreads in a very specific, smooth way, the "slope" of the hill must also follow that smoothness. The math shows that the "roughness" of the slope is limited to that $1/3$ power.

The "Injectivity" Catch

The paper admits a small limitation: This "Heat Trick" only works if the slope directions are unique (the "one-way street" rule).

Real-world check: Most simple hills (like a flat plane or a perfect cone) have slopes that repeat (many points have the same slope). So, this trick doesn't work for those specific shapes directly.
However: The paper shows that for the complex, interesting shapes where the slope is unique, the math holds up perfectly.

The Examples: Testing the Theory

The author tests his theory on two specific shapes:

The Square Hill: Imagine a hill inside a square box. In the corners, the slope is unique. The paper proves that in these corners, the slope is indeed $1/3$ -smooth. It's like finding a hidden rule that governs the "rough" edges of a square.
Aronsson's Famous Solution: This is the "sharp" hill mentioned earlier ( $x^{4/3} - y^{4/3}$ ). The paper confirms that this shape fits the theory perfectly. The "roughness" happens exactly where the slope directions line up in a specific way, and the math predicts the $1/3$ limit exactly.

Summary in One Sentence

If you have a mathematical hill where every slope direction points to a unique spot on the map, you can turn the problem into a heat equation, which proves that the slope cannot get "rougher" than a specific $1/3$ -power limit.

Why this matters (according to the paper):
It solves a long-standing guess about the maximum "roughness" allowed in these special mathematical landscapes, but only under the condition that the slopes don't repeat themselves. It uses the predictable nature of heat to tame the unpredictable nature of these geometric shapes.

Technical Summary: Infinity-harmonic functions in the plane: Regularity by injectivity

Problem Statement
The paper addresses a long-standing conjecture regarding the regularity of $\infty$ -harmonic functions in the plane ( $\mathbb{R}^2$ ). Specifically, it investigates whether the gradient of a viscosity solution to the $\infty$ -Laplace equation, $\Delta_\infty u = \nabla u \cdot D^2 u \nabla u = 0$ , possesses $1/3$ -Hölder continuity ( $C^{1, 1/3}$ ). While it is established that solutions are $C^1$ and that their gradients are locally $\alpha$ -Hölder for some $\alpha > 0$ , a uniform positive lower bound for $\alpha$ has not been proven in the general case. The paper notes that Aronsson's solution, $u(x,y) = x^{4/3} - y^{4/3}$ , demonstrates that no regularity better than $1/3$ -Hölder is possible in general.

Methodology
The author, Karl K. Brustad, establishes a connection between the $\infty$ -Laplace equation and the one-dimensional heat equation (caloric structure) under a specific geometric constraint. The core methodology involves the following steps:

Injectivity Assumption: The analysis is restricted to domains where the gradient map $\nabla u$ is injective (specifically satisfying a metric injectivity condition $|\nabla u(x) - \nabla u(y)| \ge C_1 |x-y|$ ). This ensures the existence of a Lipschitz inverse map $f: B \to \Omega$ , where $B = \nabla u(\Omega)$ .
Legendre Transform and Change of Variables: The author constructs a potential function $w(p) = p^\top f(p) - u(f(p))$ for the inverse map. A smooth, locally one-to-one change of variables is introduced, mapping the gradient space to a domain $D$ parameterized by $(\psi, t)$ , where $p = e^{-t}[\cos \psi, \sin \psi]^\top$ . Here, $e^{-t}$ represents the magnitude of the gradient and $\psi$ its direction.
Caloric Structure: It is proven that the transformed function $h(\psi, t) = w(p(\psi, t))$ satisfies the heat equation $h_{\psi\psi} = h_t$ . Consequently, the original function $u$ , when expressed in these coordinates, is also caloric.
Streamline Analysis: The proof utilizes the existence of maximal ascending and descending streamlines for $\infty$ -subharmonic and $\infty$ -superharmonic functions to analyze the behavior of the gradient along curves where its magnitude is non-decreasing.
Taylor Expansion and Nodal Analysis: To establish the regularity bound, the author performs a Taylor expansion of the caloric function $v$ (derived from $u$ ) around points where the spatial derivative $v_\psi$ vanishes. By analyzing the heat polynomials (e.g., $r_3, r_4$ ) and their nodal domains, the author demonstrates that if $v_\psi = 0$ , then $v_t \neq 0$ . This non-degeneracy is crucial for the subsequent estimates.
Inverse Function Estimates: Using the non-degeneracy of the Hessian of the potential $w$ at critical points of $v_\psi$ , the author derives a third-order expansion of the inverse map $f$ . This leads to the estimate $|f(p+h) - f(p)| \ge C|h|^3$ locally.

Key Contributions and Results
The primary result is Theorem 1.1, which states:
If $u$ is $\infty$ -harmonic in an open set $\Omega \subseteq \mathbb{R}^2$ with $\nabla u \neq 0$ , and if the gradient satisfies the metric injectivity condition $|\nabla u(x) - \nabla u(y)| \ge C_1 |x-y|$ , then locally in $\Omega$ , there exists a constant $C_2$ such that:
$|\nabla u(x) - \nabla u(y)| \le C_2 |x-y|^{1/3}$

The paper provides two specific examples to illustrate the theory:

The $\infty$ -potential in a square: The solution to the Dirichlet problem on a square with a point source is shown to have an invertible gradient in a sub-square. The associated caloric function is explicitly constructed using Jacobi Theta functions. The result confirms that the gradient is exactly $C^{1, 1/3}$ near the diagonals (where $v_\psi=0$ ) and real-analytic elsewhere.
Aronsson's solution: The classic singular solution $u = \frac{3}{4}(x^{4/3} - y^{4/3})$ is re-examined. The analysis confirms that the non-smoothness occurs precisely when the gradient direction aligns with the axes ( $v_\psi = 0$ ), and the derived regularity matches the known $1/3$ -Hölder behavior.

Significance and Claims
The paper claims to provide a proof of the $1/3$ -Hölder continuity of the gradient for $\infty$ -harmonic functions in the plane, contingent upon the injectivity of the gradient map. The author emphasizes that while injectivity is not a necessary condition (as smooth solutions like planes and cones exist), the "caloric structure" accessed via this condition is sufficient to derive the optimal regularity bound.

The author modestly notes that the proof is limited to the situation where $\nabla u \neq 0$ and does not currently include critical points, though this might be possible with further analysis of the limit $t \to \infty$ . Additionally, the author suggests that the injectivity condition might be weakened to mere injectivity (without the metric bound), but proving this would require improved versions of the supporting lemmas. The work relies on the connection between the $\infty$ -Laplace equation and the heat equation, a link previously observed by Aronsson, but formalized here to yield specific regularity estimates.