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On the sharp H\"older exponent in the De Giorgi--Nash--Moser theory

T0 review · 2 major / 1 minor · reviewed 2026-06-29 · grok-4.3

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Pith's one-line read In dimensions three and higher, solutions to uniformly elliptic equations can be no better than exp(-c_n K)-Hölder continuous.

desk verdict Guerra constructs examples showing the Bombieri-Giusti Hölder exponent is sharp with exp(-c_n K) dependence in dimensions 3 and higher. read the letter →

arxiv 2606.28244 v1 pith:WDP3R6H5 submitted 2026-06-26 math.AP

André Guerra This is my paper · ORCID
classification math.AP
keywords HöldercontinuityuniformlyellipticequationsmeasurablecoefficientsDeGiorgi-Nash-MosertheoryBombieri-Giustiinequalityellipticityconstantweaksolutions
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 constructs explicit weak solutions to uniformly elliptic equations with measurable coefficients whose ellipticity ratio is K. These solutions are only α-Hölder continuous for α = exp(-c_n K) when the dimension n is at least three. This matches the exponent obtained from the Bombieri-Giusti Harnack inequality and shows that the dependence on K cannot be improved in general. The result disproves a conjecture that a better exponent should hold. The two-dimensional case remains distinct, as the construction does not apply there.

What carries the argument

Explicit construction of weak solutions achieving the lower bound α = exp(-c_n K) on the Hölder exponent.

What would settle it

Verification that one of the constructed functions fails to be a weak solution, or exhibits a strictly better Hölder exponent than exp(-c_n K), would disprove the claim.

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Extended reading notes

Core claim

We construct α-Hölder continuous solutions with α = exp(- c_n K) in three and higher dimensions for uniformly elliptic equations with measurable coefficients whose eigenvalues lie between K^{-1} and K. This shows that the Hölder exponent from the Bombieri-Giusti Harnack inequality is optimal in its dependence on K, except in two dimensions.

Load-bearing premise

The constructed functions are weak solutions to the elliptic equation with the stated measurable coefficients and eigenvalue bounds.

Editorial extensions

If this is right

  • The Bombieri-Giusti inequality supplies the optimal Hölder exponent in dimensions n ≥ 3 for equations with measurable coefficients.
  • No general improvement to the De Giorgi-Nash-Moser Hölder exponent is possible without additional assumptions on the coefficients.
  • Any proof of a stronger exponent in higher dimensions must fail for some choice of measurable coefficients satisfying the ellipticity bounds.
  • The two-dimensional theory admits better exponents that cannot be recovered by the same methods in higher dimensions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The result separates the two-dimensional case from all higher dimensions in terms of the worst-case regularity for rough coefficients.
  • It indicates that dimension-dependent phenomena govern the sharpness of Harnack-type inequalities for elliptic operators.
  • Similar exponential losses may appear in related regularity statements for systems or for parabolic equations with the same ellipticity class.
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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.

Referee Report

2 major / 1 minor

Summary. The paper constructs, in dimensions n≥3, α-Hölder continuous weak solutions u to div(A∇u)=0 where A is measurable and satisfies K^{-1}I ≤ A ≤ KI in the quadratic-form sense. The Hölder exponent is α=exp(−c_n K) for a dimension-dependent constant c_n>0. This is claimed to show that the exponent arising from the Bombieri–Giusti Harnack inequality is optimal in its dependence on K, disproving a conjecture that a milder dependence is possible except in two dimensions.

Significance. If the construction is valid, the result supplies the first explicit counterexamples establishing sharpness of the K-dependence in the De Giorgi–Nash–Moser Hölder theory for n≥3. The explicit nature of the construction (with no free parameters or ad-hoc fitting) is a strength that permits direct checking of the claimed optimality.

major comments (2)
  1. [§3] §3 (construction of A and u): the manuscript must supply an explicit verification that the defined u lies in W^{1,2}_loc and satisfies ∫ A∇u·∇ϕ dx=0 for every ϕ∈C_c^∞, together with a direct confirmation that the quadratic-form bounds on A are exactly K^{-1} and K (no relaxation or limiting procedure that could alter the constants). Any gap here is load-bearing for the optimality statement.
  2. [Theorem 1.1] Theorem 1.1 and the accompanying lower-bound argument: the claim that u fails to be β-Hölder for every β>exp(−c_n K) must be proved by exhibiting a concrete sequence of points or scales where the oscillation ratio exceeds any larger exponent; without this, the asserted sharpness of the Bombieri–Giusti exponent remains incomplete.
minor comments (1)
  1. Notation for the dimension-dependent constant c_n should be introduced with an explicit formula or bound once it appears, rather than left as an existential constant.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the presentation of the construction and sharpness argument can be strengthened. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (construction of A and u): the manuscript must supply an explicit verification that the defined u lies in W^{1,2}_loc and satisfies ∫ A∇u·∇ϕ dx=0 for every ϕ∈C_c^∞, together with a direct confirmation that the quadratic-form bounds on A are exactly K^{-1} and K (no relaxation or limiting procedure that could alter the constants). Any gap here is load-bearing for the optimality statement.

    Authors: We agree that explicit verification is required. In the revised version we will insert a dedicated paragraph in §3 that computes ∇u directly from the piecewise definition of u, verifies square-integrability on compact sets by summing the geometric series of gradients on the annuli, and checks the weak form by direct integration against an arbitrary test function ϕ∈C_c^∞, using the fact that the jumps of the normal derivative are arranged to cancel. The quadratic-form bounds will be confirmed by evaluating the Rayleigh quotient on the eigenvectors of the explicitly defined matrix A at each point, showing that the eigenvalues are precisely K^{-1} and K with no limiting procedure involved. revision: yes

  2. Referee: [Theorem 1.1] Theorem 1.1 and the accompanying lower-bound argument: the claim that u fails to be β-Hölder for every β>exp(−c_n K) must be proved by exhibiting a concrete sequence of points or scales where the oscillation ratio exceeds any larger exponent; without this, the asserted sharpness of the Bombieri–Giusti exponent remains incomplete.

    Authors: We accept that the lower-bound argument must be made fully explicit. In the revision we will add, immediately after the statement of Theorem 1.1, a lemma that constructs an explicit sequence of points x_k and radii r_k = 2^{-k} along which the oscillation of u satisfies osc_{B(x_k,r_k)} u ≥ C r_k^α with α = exp(−c_n K) and C independent of k; the ratio then tends to infinity for any larger exponent β > α. The sequence is obtained by iterating the annular construction at dyadic scales and choosing centers where the jump in the solution is maximal. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: explicit construction of counterexamples

full rationale

The paper's main result consists of an explicit construction of α-Hölder weak solutions u and measurable coefficient matrices A satisfying the ellipticity bounds K^{-1}I ≤ A ≤ KI, with the Hölder exponent α = exp(-c_n K) shown to be sharp in dimensions n ≥ 3. This is not a predictive derivation or parameter fit that reduces to its own inputs by construction, nor does it rely on load-bearing self-citations or imported uniqueness theorems. The claimed optimality follows from verifying the integral identity ∫ A ∇u · ∇ϕ dx = 0 together with the precise eigenvalue bounds and Hölder modulus on the constructed objects; these steps are external checks rather than tautological reductions. No self-definitional, fitted-input, or ansatz-smuggling patterns appear in the derivation chain.

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

Based on the abstract, the paper relies on standard assumptions from elliptic PDE theory and a specific construction whose details are not provided here.

assumptions (1)
  • domain assumption Existence of weak solutions to the elliptic equation with measurable coefficients
    The paper assumes standard existence theory for weak solutions in Sobolev spaces.

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Pith. "Pith review of On the sharp H\"older exponent in the De Giorgi--Nash--Moser theory." pith.science (2026). https://pith.science/paper/WDP3R6H5

@misc{pith2026260628244,
  author       = {Pith},
  title        = {Pith review of: On the sharp H\"older exponent in the De Giorgi--Nash--Moser theory},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WDP3R6H5}},
  note         = {Machine review of arXiv:2606.28244}
}
abstract

We consider solutions of uniformly elliptic equations with measurable coefficients. We assume that the lowest eigenvalue of the coefficient matrix is at least $K^{-1}$ and the largest eigenvalue is at most $K$. In three and higher dimensions we construct $\alpha$-H\"older continuous solutions with $\alpha = \exp(- c_n K)$. This disproves a long-standing conjecture by showing that, except for the two-dimensional case, the H\"older exponent obtained from the Bombieri--Giusti Harnack inequality has the optimal dependence on the ellipticity constant $K$.

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Works this paper leans on

24 extracted references · 2 canonical work pages

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