Infinity-harmonic functions in the plane: Regularity by injectivity

Karl K. Brustad

Infinity-harmonic functions in the plane: Regularity by injectivity

T0 review · 0 major / 2 minor · reviewed 2026-06-27 · grok-4.3

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Pith's one-line read Under an injectivity condition on the gradient, infinity-harmonic functions in the plane have 1/3-Hölder continuous gradients.

desk verdict This paper gives a conditional proof of 1/3-Hölder gradient regularity for plane infinity-harmonic functions by using injectivity to reach the 1D heat equation structure. read the letter →

arxiv 2606.08257 v1 pith:TVVKJL73 submitted 2026-06-06 math.AP

Karl K. Brustad This is my paper · ORCID

classification math.AP

keywords infinity-harmonicfunctionsgradientregularityinjectivityconditionHoldercontinuityinfinity-Laplaceequationone-dimensionalheatplane

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper shows that infinity-harmonic functions in the plane whose gradient satisfies an injectivity condition have gradients that are 1/3-Hölder continuous. This follows from accessing the known link to the one-dimensional heat equation and using its caloric structure to obtain the regularity. The result targets a long-standing conjecture on the optimal regularity of these functions, which are already known to be C^1 but lack a uniform positive Hölder exponent on the gradient in general. A sympathetic reader cares because the condition isolates a class where the conjectured exponent is achieved, even though the condition is sufficient but not necessary.

What carries the argument

The injectivity condition on the gradient, which grants access to the caloric structure from the connection to the one-dimensional heat equation.

What would settle it

An infinity-harmonic function in the plane whose gradient is injective but fails to be 1/3-Hölder continuous would falsify the claim.

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Extended reading notes

Core claim

Under a certain injectivity condition on the gradient, the link between the infinity-Laplace equation and the one-dimensional heat equation becomes accessible, and the resulting caloric structure proves the 1/3-Hölder continuity of the gradient for infinity-harmonic functions in the plane.

Load-bearing premise

The gradient of the infinity-harmonic function must satisfy the injectivity condition to access the caloric structure from the heat equation link.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is a conditional result: under an injectivity condition on the gradient, infinity-harmonic functions in the plane have 1/3-Hölder continuous gradients, obtained by accessing the one-dimensional heat equation structure.

This is new because it provides a concrete way to use Aronsson's observed connection between the infinity-Laplace equation and the heat equation to prove the sharp regularity. The paper shows how the caloric properties yield the Hölder bound once injectivity is assumed.

The paper does well in being upfront about the limitations. It states clearly that the injectivity condition is sufficient but not necessary, and it gives counterexamples like planes and cones where the gradient is not injective but the functions are still smooth.

The soft spots are mostly around the conditional nature. This does not solve the general conjecture, only a special case. The derivation from injectivity to the heat equation and then to the 1/3-Hölder is not spelled out in the abstract, so the strength depends on whether those steps are rigorous and free of additional assumptions in the full text. No signs of circularity or fitting appear in the description.

Overall, the argument seems to engage honestly with the literature and the known sharpness example.

This work is for experts in nonlinear PDEs and viscosity solutions who follow the infinity-harmonic regularity problem. A reader interested in new techniques for conditional regularity would get something from it.

I think it should go to peer review. The idea is worth having referees examine the details of how the injectivity unlocks the caloric structure.

Referee Report

0 major / 2 minor

Summary. The manuscript claims that under a sufficient (but not necessary) injectivity condition on the gradient, the known link between the infinity-Laplace equation and the one-dimensional heat equation in the plane becomes accessible, and the resulting caloric structure yields the sharp 1/3-Hölder continuity of the gradient for infinity-harmonic functions. The paper recalls the known C^1 regularity and local α-Hölder continuity (with no uniform positive lower bound on α), notes that Aronsson's example x^{4/3}-y^{4/3} shows 1/3 is optimal, and supplies counter-examples (planes, cones) where the injectivity condition fails.

Significance. If the derivation holds, the result supplies a conditional route to the long-standing 1/3-Hölder conjecture for the gradient of plane infinity-harmonic functions by converting the injectivity assumption into access to the heat equation. The approach is proportionate to the known sharpness example and correctly flags that the condition is sufficient rather than necessary. The manuscript gives explicit credit to prior C^1 results and to Aronsson's original observation of the caloric link.

minor comments (2)

[Abstract] Abstract: the phrase 'a certain injectivity condition' is used without a forward reference to its precise definition or to the section in which it is introduced; adding a parenthetical pointer would improve readability for readers who stop at the abstract.
The manuscript could usefully include a short remark, perhaps in the introduction, on whether the injectivity condition is expected to hold for a dense class of solutions or only for a special subclass.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No major comments were raised in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity

full rationale

The central claim is explicitly conditional on an injectivity assumption for the gradient that grants access to the 1D heat equation structure. This assumption is stated as sufficient but not necessary, with counter-examples (planes, cones) provided where it fails. No load-bearing step reduces by construction to a fitted input, self-definition, or self-citation chain; the derivation is self-contained against the stated hypothesis and matches independent sharpness examples.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

The argument rests on the pre-existing connection between the infinity-Laplace equation and the one-dimensional heat equation (observed by Aronsson) together with the new injectivity hypothesis; no free parameters or invented entities are mentioned.

assumptions (1)

domain assumption The known link between infinity-harmonic functions in the plane and the one-dimensional heat equation holds.
Invoked in the abstract as the structure that becomes usable under injectivity.

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Cite this review

Pith. "Pith review of Infinity-harmonic functions in the plane: Regularity by injectivity." pith.science (2026). https://pith.science/paper/TVVKJL73

@misc{pith2026260608257,
  author       = {Pith},
  title        = {Pith review of: Infinity-harmonic functions in the plane: Regularity by injectivity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TVVKJL73}},
  note         = {Machine review of arXiv:2606.08257}
}

abstract

It has been a long standing conjecture that the $ \infty $-harmonic functions in the plane have a 1/3-H\"older continuous gradient. It \emph{is} known that solutions are $ C^1 $ and that the gradient is locally $ \alpha $-H\"older, but $ \alpha $ comes without any positive lower bound. Aronsson's solution $ x^{4/3} - y^{4/3} $ shows that no better general regularity is possible. In the plane there is also a connection between the $ \infty $-Laplace equation and the one-dimensional heat equation, observed already by Aronsson himself. I shall show that this link can be accessed under a certain injectivity condition on the gradient, and that the caloric structure then is enough to prove the 1/3-H\"older continuity. Of course, an injective gradient is by no means a \emph{necessary} condition, as seen by smooth solutions such as the planes and cones.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Infinity-harmonic functions and inverse mean curvature flow clusters
math.AP 2026-07 accept novelty 8.0 of 10

Infinity-harmonic functions on planar domains are C^{1,1/3} with isolated critical points and unique quasiradial blow-ups, via a p-to-infinity duality that produces inverse mean curvature flow clusters.

Reference graph

Works this paper leans on

3 extracted references · cited by 1 Pith paper

[1]

On the number of nodal domains of homogeneous caloric polynomials

Matthew Badger and Cole Jeznach. On the number of nodal domains of homogeneous caloric polynomials. Annales Henri Lebesgue , 9:407--437, 2026

2026
[2]

Karl K. Brustad. The infinity-potential in the square. Advances in Calculus of Variations , 19(1):43--59, 2026

2026
[3]

Crandall

Michael G. Crandall. A visit with the - Laplace equation. In Calculus of Variations and Nonlinear Partial Differential Equations , volume 1927 of Lecture Notes in Mathematics , pages 75--122. Springer, Berlin, Heidelberg, 2008

1927

Pith tools

Reviewed June 27, 2026 · model on record in the stance chip above.

[1] [1]

On the number of nodal domains of homogeneous caloric polynomials

Matthew Badger and Cole Jeznach. On the number of nodal domains of homogeneous caloric polynomials. Annales Henri Lebesgue , 9:407--437, 2026

2026

[2] [2]

Karl K. Brustad. The infinity-potential in the square. Advances in Calculus of Variations , 19(1):43--59, 2026

2026

[3] [3]

Crandall

Michael G. Crandall. A visit with the - Laplace equation. In Calculus of Variations and Nonlinear Partial Differential Equations , volume 1927 of Lecture Notes in Mathematics , pages 75--122. Springer, Berlin, Heidelberg, 2008

1927