Uniqueness of solutions to an elliptic inequality with rapid decay at infinity

F. Golgeleyen; M. Yamamoto; O.Y. Imanuvilov

Uniqueness of solutions to an elliptic inequality with rapid decay at infinity

T0 review · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A Carleman estimate with a weight depending only on y shows that anisotropic exponential/polynomial decay forces u=0 for elliptic inequalities in exterior domains.

desk verdict A new anisotropic decay uniqueness theorem with a genuinely proven Carleman estimate, but a repairable polynomial-weight error in the proof that the referee should catch. read the letter →

arxiv 2505.14431 v1 pith:GCPILL6U submitted 2025-05-20 math.AP

F. Golgeleyen , O.Y. Imanuvilov , M. Yamamoto This is my paper · ORCID

classification math.AP

keywords vertgammathetaconstantscoordinatesdecaydomainelliptic

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 studies solutions u of an elliptic inequality, |Δu| ≤ C0(|y|^{-γ}|u| + |y|^{-θ}|∇u|), outside a bounded obstacle U. Here x = (y,z), with y an m-dimensional coordinate block and z the remaining coordinates. The authors ask: if u decays fast in the y variables, say like exp(-C1|y|^{2β}), and decays only polynomially in the z variables, must u be zero everywhere?

This is in the tradition of Landis' conjecture, where the question is how fast a nontrivial solution of an elliptic equation can decay at infinity. Meshkov showed the critical exponential rate is 4/3: decay faster than exp(-|x|^{4/3+ε}) forces zero, and this is sharp. The current paper transfers this idea to anisotropic decay. The main theorem says that if β is large enough, specifically larger than a constant α derived from the exponents γ and θ, then the only solution is u=0.

The technical core is a Carleman estimate, a weighted integral inequality. The new twist is that the weight function depends only on the y coordinates, φ(x)=|y|^{2α}, not on all variables. This allows the exponential decay in y to be exploited without requiring any exponential decay in z. The proof also uses standard cut-off functions to localize the problem near infinity.

A separate section gives simpler uniqueness results for equations with nonnegative potential V and drift A satisfying div A ≥ 0: if the average of u^2 over spherical shells tends to zero (at a suitable rate), then u=0. These require no decay of the gradient at all.

Extended reading notes

Core claim

The load-bearing assertion is Theorem 1.1: 'We assume that u ∈ H^2_loc(R^n \ U) satisfies (1.4) and there exist constants C>0, C1>0 and β>α such that |u(x)| ≤ C g(|z|) exp(-C1 |y|^{2β}), x ∈ R^n \ U. Then u = 0 in R^n \ U.' If the paper is correct, anisotropic fast decay in the y-variables plus polynomial decay in z forces the trivial solution.

Load-bearing premise

The strong unique continuation property for the inequality (1.4) in the exterior domain, with coefficients |y|^{-γ} and |y|^{-θ} that may be singular near y=0 when γ or θ is positive. The proof establishes u=0 on {|y|≥7}×R^{n-m} and then invokes unique continuation (Choulli [2]) to conclude u=0 in all of R^n\U. This cited property, not proved in the paper, is load-bearing for the final conclusion.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Dear Colleague,

Read this paper. It proves a new uniqueness theorem for the elliptic inequality |Δu| ≤ C0(|y|^{-γ}|u| + |y|^{-θ}|∇u|) in an exterior domain, under exponential decay in m of the variables and polynomial decay in the rest. The anisotropic decay condition (1.7) and the Carleman weight |y|^{2α} are new as far as I can tell; the isotropic case m=n recovers known results but the partial-coordinate version is the contribution. The Carleman estimate in Section 5 is fully proven, self-contained, and the absorption conditions defining α come from the inequality itself, so there's no circularity.

The main theorem is likely correct, but the written proof contains a concrete error in the estimation of S1 (and similarly S2, S3). Around (2.14), the authors bound |y|^{σ1}q_R^1 by C s^2 |y|^{σ2} with σ2 = σ1 + 4α. Lemma 2.4 gives |Δq| + |q| ≤ C s^2 |y|^{σ1+4α} e^{2sφ}, but the second term in Lemma 2.2 is |y|^{σ1}q, which is |y|^{2σ1} e^{2sφ}. When σ1 > 4α (which happens, e.g., for γ < -4 or large |θ|), 2σ1 > σ1+4α and the displayed inequality is false as written. This is repairable: taking σ2 = max{σ1+4α, 2σ1} restores the estimate, and the limits as R→∞ still vanish because β > α dominates any polynomial. So the theorem survives, but the submitted proof needs a correction.

Other notes: Theorem 1.4 is sketched in a few lines, relying entirely on the same argument; the unique continuation step from the unbounded strip to all of R^n\U is imported from Choulli [2] without verifying its hypotheses here. These are legitimate soft spots but not fatal.

The paper deserves a serious referee. It is a genuine, checkable contribution to the Landis and unique continuation literature. I'd recommend engaging with it, and asking the authors to fix the σ2 definition and expand the proof of Theorem 1.4.

Best

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The proof is self-contained apart from the standard strong unique continuation property and basic Sobolev space tools. No new physical or geometric entities are postulated.

assumptions (3)

domain assumption Strong unique continuation for elliptic inequalities with coefficients |y|^{-γ}, |y|^{-θ}
Invoked in Section 2 after proving u=0 in {|y|≥7}×R^{n-m} to conclude u=0 in R^n\U; cited to Choulli [2]. The coefficients can be singular near y=0 for γ,θ>0, and no proof is provided in this paper.
domain assumption U simply connected bounded domain implies the component of R^n\U connected with infinity is all of R^n\U
Stated in Section 2: 'Since U is simply connected, such a connected component is nothing but R^n\U.'
standard math Standard integration by parts and Green's formula for H^2 functions
Used throughout the proofs of Lemmas 2.2, 5.1, and Theorems 4.1, 4.3.

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Pith. "Pith review of Uniqueness of solutions to an elliptic inequality with rapid decay at infinity." pith.science (2026). https://pith.science/paper/GCPILL6U

@misc{pith2026250514431,
  author       = {Pith},
  title        = {Pith review of: Uniqueness of solutions to an elliptic inequality with rapid decay at infinity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GCPILL6U}},
  note         = {Machine review of arXiv:2505.14431}
}

abstract

We consider an elliptic differential inequality: $\vert \Delta u(x) \vert \le C_0(\YYYY^{-\gamma}\vert u(x)\vert + \YYYY^{-\theta}\vert \nabla u(x)\vert)$ in an exterior domain $\R^n \setminus \ooo{U}$, where $U$ is a simply connected bounded domain $U$, $x := (y,z) \in \R^n$ with $y \in \R^m$ and $z\in \R^{n-m}$ for given $m\in \{ 1, ..., n\}$, and $\gamma, \theta \in \R$ are constants. We assume that $u(x)$ decays with exponential rate in the $y$-coordinates and polynomial rate in the $z$-coordinates as $\vert x\vert \to \infty$. We prove that if decay rates of $u$ satisfy certain conditions related to the constants $\gamma, \theta \in \R$, then $u\equiv 0$ in $\UUUUU$. The key is a Carleman estimate with typical cut-off arguments.

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Works this paper leans on

21 extracted references · 20 canonical work pages

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [2]

Choulli, Applications of Elliptic Carleman Inequalities to Caucy an d Inverse Problems , Springer Nature, Berlin, 2016

M. Choulli, Applications of Elliptic Carleman Inequalities to Caucy an d Inverse Problems , Springer Nature, Berlin, 2016

work page 2016

[2] [1]

Alessandrini, Strong unique continuation for general elliptic eq uations in 2D, J

G. Alessandrini, Strong unique continuation for general elliptic eq uations in 2D, J. Math. Anal. App. 386 (2012) 669-676

work page 2012

[3] [3]

Das and Y

U. Das and Y. Pinchover, The Landis conjecture via Liouville compa rison pricniple and criticality theory, preprint, 2024, arXiv:2405.11695

arXiv 2024

[4] [4]

Davey, Some quantitative unique continuation results for eige nfunctions of the magnetic Schr¨ odinger operator, Comm

B. Davey, Some quantitative unique continuation results for eige nfunctions of the magnetic Schr¨ odinger operator, Comm. Partial Diﬀ. Eqs. 39 (2014) 876-945. 30 1 F. G ¨OLGELEYEN, 2 O.Y. IMANUVILOV AND 3 M. YAMAMOTO

work page 2014

[5] [5]

Davey, On Landis’ conjecture in the plane for some equations with sign-changing potentials, Rev

B. Davey, On Landis’ conjecture in the plane for some equations with sign-changing potentials, Rev. Mat. Iberoam. 36 (2020) 1571-1596

work page 2020

[6] [6]

Davey, On Landis’ conjecture in the plane for potentials with g rowth, Vietnam Journal of Mathe- matics 52 (2024) 675-688

B. Davey, On Landis’ conjecture in the plane for potentials with g rowth, Vietnam Journal of Mathe- matics 52 (2024) 675-688

work page 2024

[7] [7]

Davey, C

B. Davey, C. Kenig and J.-N. Wang, The Landis conjecture for va riable coeﬃcient second-order elliptic PDEs, Trans. Amer. Math. Soc. 369 (2017) 8209-8237

work page 2017

[8] [8]

Davey, C

B. Davey, C. Kenig and J.-N. Wang, On Landis’ conjecture in the p lane when the potential has an exponentially decaying negative part, St. Petersburg Math. J. 31 (2020) No. 2, 337-353

work page 2020