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arxiv: 2604.13303 · v1 · submitted 2026-04-14 · 🧮 math.AP

Harnack inequality for non-uniformly elliptic equations in non-divergence form

David Bowman This is my paper

Pith reviewed 2026-05-10 14:09 UTC · model grok-4.3

classification 🧮 math.AP
keywords degenerate elliptic equationsHarnack inequalitynon-divergence formLiouville theoremweak Harnack inequalityoscillation estimateslogarithmic maximum principle
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The pith

For elliptic equations with integrable degeneracy, Harnack inequalities hold when the integrability exponent exceeds a dimension-dependent threshold.

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

The paper establishes that solutions to the non-divergence elliptic equation a_ij partial_ij u = 0 satisfy improved oscillation estimates and a Liouville theorem whenever the lower ellipticity bound λ obeys λ^{-1} in L^p for p greater than the dimension minus one. A stronger Harnack inequality follows once p is taken sufficiently large depending on dimension. These conclusions rest on a new logarithmic weak Harnack inequality for supersolutions and a logarithmic local maximum principle obtained by touching subsolutions with double-exponential barriers; both auxiliary results hold for p > d-1 and the maximum principle is new even without degeneracy. Counterexamples demonstrate that Harnack and weak Harnack inequalities fail when p drops below d-1 and that power-type integrability estimates cannot hold for any finite p.

Core claim

Under the sole assumption I ≥ a_ij(x) ≥ λ(x) I with λ^{-1} ∈ L^p, the equation a_ij partial_ij u = 0 admits an improvement of oscillation for every p > d-1, which immediately yields a Liouville theorem for bounded solutions. When p is large enough relative to dimension a Harnack inequality holds. The proofs proceed by first deriving a new log-L^ε weak Harnack inequality for supersolutions, then using double-exponential blow-up barriers to touch subsolutions and obtain a logarithmic local maximum principle; both of these hold already for p > d-1 and the local maximum principle is new even in the uniformly elliptic case. Sharpness is shown by explicit examples: no Harnack or weak Harnack is p

What carries the argument

The log-L^ε weak Harnack inequality for supersolutions combined with double-exponential blow-up barriers that produce a logarithmic local maximum principle for subsolutions.

If this is right

  • Bounded solutions are necessarily constant when p > d-1.
  • Positive solutions satisfy a Harnack inequality once p is large enough depending on dimension.
  • A logarithmic local maximum principle holds even for uniformly elliptic equations.
  • No Harnack or weak Harnack inequality is possible when p < d-1, and power-type L^ε estimates fail for every finite p.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Logarithmic rather than power-type estimates may extend to other degenerate or singular operators where power integrability breaks down.
  • The barrier construction could be adapted to test similar results for parabolic equations or fully nonlinear operators under the same integrability assumption on λ.
  • The threshold p = d-1 may relate to capacity or Sobolev embedding phenomena that govern removable singularities for these degenerate operators.

Load-bearing premise

The only control on the coefficients is that their lower ellipticity bound λ satisfies λ^{-1} ∈ L^p for p large enough relative to dimension.

What would settle it

An explicit non-constant bounded solution (or a positive solution whose maximum-to-minimum ratio is arbitrarily large inside a ball) for some λ with λ^{-1} ∈ L^p when p exceeds d-1 would disprove the oscillation improvement, Liouville, or Harnack claims.

Figures

Figures reproduced from arXiv: 2604.13303 by David Bowman.

Figure 1
Figure 1. Figure 1: The construction of the inner balls B. Note that the outer ball Be would extend far outside of B1. As a consequence, we may apply the pigeonhole principle to select a subcover Ci∗ such that [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
read the original abstract

We study regularity properties for solutions to the nakedly degenerate elliptic equation $a_{ij}\partial_{ij}u =0$, where the coefficients satisfy $I \ge a_{ij}(x) \ge \lambda(x) I$ and the only assumption is that $\lambda^{-1} \in L^p$. We prove an improvement of oscillation and a Liouville theorem for $p>d-1$, and a Harnack inequality for $p$ sufficiently large depending on dimension. Along the way, we obtain a new $\log-L^\varepsilon$ Weak Harnack inequality for supersolutions. Then, touching subsolutions by double exponential blow-up barriers, we also derive a logarithmic local maximum principle that is new even in the uniformly elliptic case. Both of these results hold for $p>d-1$. Finally, we construct examples showing that there cannot be Harnack or Weak Harnack inequalities in the regime $p<d-1$, nor can there be power-type $L^\varepsilon$ inequalities in the case of any $p<\infty$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies regularity for solutions of the non-divergence equation a_{ij} ∂_{ij} u = 0 where the coefficients satisfy I ≥ a_{ij}(x) ≥ λ(x) I with the sole structural assumption λ^{-1} ∈ L^p. It establishes an improvement of oscillation and a Liouville theorem when p > d-1, a Harnack inequality when p is sufficiently large (dimension-dependent), a new log-L^ε weak Harnack inequality for supersolutions, and a logarithmic local maximum principle obtained via double-exponential blow-up barriers (new even for uniformly elliptic operators). Both auxiliary results hold for p > d-1. Sharpness is shown by counterexamples: no Harnack or weak Harnack inequalities exist for p < d-1, and no power-type L^ε inequalities hold for any finite p.

Significance. If the derivations hold, the results are significant: they give sharp integrability thresholds on the degeneracy for oscillation and Harnack-type controls in the non-divergence setting, introduce a logarithmic local maximum principle that improves even the uniformly elliptic theory, and supply matching counterexamples. The double-exponential barrier technique and the log-L^ε weak Harnack are technically novel contributions.

major comments (2)
  1. [Section on logarithmic local maximum principle] Proof of the logarithmic local maximum principle (via double-exponential barriers): the construction must be verified to remain valid under the sole assumption λ^{-1} ∈ L^p for p > d-1; the dependence of the constants on p and on the L^p norm of λ^{-1} should be tracked explicitly, as the barrier height grows double-exponentially and may interact with the integrability.
  2. [Counterexamples] Counterexample section for p < d-1: the constructed coefficients and solutions must be checked to satisfy the equation in the viscosity (or appropriate weak) sense while keeping λ^{-1} exactly in L^p but not better; the argument that no power-type L^ε inequality can hold for any finite p also requires an explicit verification that the example does not accidentally satisfy a stronger integrability condition.
minor comments (2)
  1. [Main theorems] The statement of the main Harnack inequality should make the dimension-dependent lower bound on p fully explicit (or at least give the functional dependence on d) rather than leaving it as 'sufficiently large'.
  2. [Weak Harnack section] Notation for the log-L^ε weak Harnack inequality: clarify whether the exponent ε depends on p and d or is universal, and state the precise form of the inequality (including the measure of the set where the log is controlled).

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and will incorporate the requested verifications and clarifications into the revised manuscript.

read point-by-point responses

  1. Referee: [Section on logarithmic local maximum principle] Proof of the logarithmic local maximum principle (via double-exponential barriers): the construction must be verified to remain valid under the sole assumption λ^{-1} ∈ L^p for p > d-1; the dependence of the constants on p and on the L^p norm of λ^{-1} should be tracked explicitly, as the barrier height grows double-exponentially and may interact with the integrability.

    Authors: We appreciate the referee's suggestion to strengthen the presentation. The double-exponential barrier construction is applied after the oscillation improvement, which holds under the sole assumption that λ^{-1} ∈ L^p for p > d-1. In the revised version we will explicitly track the dependence of all constants on p and on ||λ^{-1}||_{L^p}, and we will add a short remark (or appendix paragraph) confirming that the choice of barrier parameters depends only on these quantities and does not require any stronger integrability. revision: yes

  2. Referee: [Counterexamples] Counterexample section for p < d-1: the constructed coefficients and solutions must be checked to satisfy the equation in the viscosity (or appropriate weak) sense while keeping λ^{-1} exactly in L^p but not better; the argument that no power-type L^ε inequality can hold for any finite p also requires an explicit verification that the example does not accidentally satisfy a stronger integrability condition.

    Authors: We agree that additional explicit checks will improve clarity. In the revision we will include a detailed verification that the constructed coefficients and solutions satisfy the equation in the viscosity sense. We will also compute the precise integrability of λ^{-1} to confirm membership in L^p but not in L^q for any q > p, and we will verify directly that the solution fails every power-type L^ε inequality. These computations will be added to the counterexample section. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper establishes oscillation improvement, Liouville theorems, log-L^ε weak Harnack inequalities, and Harnack inequalities for the non-divergence equation a_ij u_ij = 0 under the sole assumption λ^{-1} ∈ L^p (p > d-1 or dimension-dependent larger p). These follow from direct analytic arguments: adapted oscillation lemmas, double-exponential barrier constructions for the logarithmic maximum principle, and explicit counterexamples showing sharpness for p < d-1. No step reduces a claimed result to a fitted parameter, self-definition, or load-bearing self-citation; the proofs are self-contained against the stated integrability hypothesis and do not invoke prior results by the same author as an unverified uniqueness theorem. This is the expected outcome for a pure existence/regularity argument in elliptic PDE theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard elliptic PDE assumptions in non-divergence form together with the given integrability condition on the degeneracy; no free parameters or new entities are introduced.

axioms (1)
  • domain assumption The equation a_ij ∂_ij u = 0 with I ≥ a_ij(x) ≥ λ(x) I where λ^{-1} ∈ L^p
    This is the precise setup of the problem studied.

pith-pipeline@v0.9.0 · 5479 in / 1029 out tokens · 38028 ms · 2026-05-10T14:09:54.972982+00:00 · methodology

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Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

  1. [1]

    Albritton and H

    D. Albritton and H. Dong. Regularity properties of passive scalars with rough divergence-free drifts.Archive for Rational Mechanics and Analysis, 247:44, 2023

    work page 2023

  2. [2]

    S. N. Armstrong, L. E. Silvestre, and C. K. Smart. Partial regularity of solutions of fully nonlinear, uniformly elliptic equations.Communications on Pure and Applied Mathematics, 65(8):1169–1184, 2012

    work page 2012

  3. [3]

    S. N. Armstrong and C. K. Smart. Regularity and stochastic homogenization of fully nonlinear equations without uniform ellipticity.Annals of Probability, 42(6):2558–2594, 2014

    work page 2014

  4. [4]

    Bella and M

    P. Bella and M. Schäffner. Local boundedness and harnack inequality for solutions of linear nonuniformly elliptic equations. Communications on Pure and Applied Mathematics, 74(4):453–477, 2021

    work page 2021

  5. [5]

    L. A. Caffarelli and X. Cabré.Fully Nonlinear Elliptic Equations, volume 43 ofColloquium Publications. American Math- ematical Society, 1995

    work page 1995

  6. [6]

    L. A. Caffarelli, M. G. Crandall, M. Kocan, and A. Świe. On viscosity solutions of fully nonlinear equations with measurable ingredients.Communications on Pure and Applied Mathematics, 49(4):365–397, 1996

    work page 1996

  7. [7]

    L. A. Caffarelli and C. E. Gutiérrez. Properties of the solutions of the linearized monge-ampère equation.American Journal of Mathematics, 119(2):423–465, 1997

    work page 1997

  8. [8]

    E. B. Fabes, C. E. Kenig, and R. P. Serapioni. The local regularity of solutions of degenerate elliptic equations.Commu- nications in Partial Differential Equations, 7(1):77–116, 1982

    work page 1982

  9. [9]

    Franchi, R

    B. Franchi, R. Serapioni, and F. Cassano. Irregular solutions of linear degenerate elliptic equations.Potential Anal., 9(3):201–216, 1998

    work page 1998

  10. [10]

    Gilbarg and N

    D. Gilbarg and N. S. Trudinger.Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, 2 edition, 2001

    work page 2001

  11. [11]

    Ignatova, I

    M. Ignatova, I. Kukavica, and L. Ryzhik. The harnack inequality for second-order elliptic equations with divergence-free drifts.Communications in Mathematical Sciences, 12(4):681–694, 2014

    work page 2014

  12. [12]

    Imbert and L

    C. Imbert and L. Silvestre. Estimates on elliptic equations that hold only where the gradient is large.Journal of the European Mathematical Society, 18(6):1321–1338, 2016

    work page 2016

  13. [13]

    Kim and S.-C

    J. Kim and S.-C. Lee. Regularity for solutions of non-uniformly elliptic equations in non-divergence form.Revista Matemática Iberoamericana, 41(6):2335–2356, 2025

    work page 2025

  14. [14]

    C. Mooney. Harnack inequality for degenerate and singular elliptic equations with unbounded drift.Journal of Differential Equations, 258(5):1577–1591, 2015

    work page 2015

  15. [15]

    O. Savin. Small perturbation solutions for elliptic equations.Communications in Partial Differential Equations, 30(1– 3):557–578, 2005

    work page 2005

  16. [16]

    N. S. Trudinger. On the regularity of generalized solutions of linear, non-uniformly elliptic equations.Archive for Rational Mechanics and Analysis, 42:50–62, 1971. (David Bowman)Department of Mathematics, University of Chicago, Chicago, Illinois 60637, USA Email address:dbowman@uchicago.edu

    work page 1971