Optimal Asymptotic Behavior at Infinity of Solutions to the Lagrangian Mean Curvature Equation with Supercritical Phase in Dimension Two

Jiguang Bao; Qinfeng Jiang

arxiv: 2604.26246 · v1 · submitted 2026-04-29 · 🧮 math.AP

Optimal Asymptotic Behavior at Infinity of Solutions to the Lagrangian Mean Curvature Equation with Supercritical Phase in Dimension Two

Jiguang Bao , Qinfeng Jiang This is my paper

Pith reviewed 2026-05-07 13:31 UTC · model grok-4.3

classification 🧮 math.AP

keywords Lagrangian mean curvature equationasymptotic behavior at infinityexterior domainssupercritical phasenonlocal methodLipschitz perturbationoptimal decay rates

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The pith

Solutions to the supercritical Lagrangian mean curvature equation in two dimensions converge to quadratic polynomials at infinity under merely Lipschitz perturbations that decay at any positive rate.

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

The paper establishes optimal asymptotic behavior at infinity for solutions of the equation arctan λ1(D²u) + arctan λ2(D²u) = θ + f(x) on exterior domains in R². It shows this holds when f is Lipschitz continuous and satisfies f(x) = O(|x|^{-β}) for any β > 0, extending earlier results that needed at least C³ regularity and β > 2. A sympathetic reader would care because the result gives precise control over the large-scale shape of the solution for a much broader class of forcing terms. The authors achieve this via a nonlocal method that avoids higher regularity assumptions on f.

Core claim

For constant phase θ with |θ| in (0, π) and Lipschitz perturbation f decaying like |x|^{-β} with β > 0, every solution u on an exterior domain in R² admits an asymptotic expansion at infinity to a quadratic polynomial satisfying the homogeneous equation, and the error rates in this expansion are optimal in the sense that they cannot be improved for the given decay of f.

What carries the argument

The nonlocal method applied to the linearized operator around the candidate quadratic asymptotic limit, which converts the decay information on f into pointwise estimates on the difference u minus the quadratic.

If this is right

The error between the solution and its quadratic asymptotic limit is controlled by the decay exponent β of f.
Convergence holds on every exterior domain once the decay condition is met.
The C³ and β > 2 requirements from prior work are unnecessary.
The same rates remain sharp for the enlarged class of Lipschitz perturbations.

Where Pith is reading between the lines

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

The result suggests that the set of entire solutions in the plane is essentially parameterized by quadratics plus a remainder whose size is dictated by the decay of f.
Analogous nonlocal arguments could be tested for the same equation in dimensions greater than two under comparable decay assumptions.
The optimality statements invite construction of explicit examples that saturate the predicted rates for each β.

Load-bearing premise

The perturbation f is Lipschitz continuous and decays at infinity at least as fast as the power |x| to the negative β for some β > 0.

What would settle it

An explicit solution on an exterior domain whose difference from any quadratic polynomial decays slower than the rate predicted by the value of β, or fails to converge at all when f is Lipschitz but β = 0.

read the original abstract

We employ a nonlocal method to study the asymptotic behavior at infinity ofsolutions to the two-dimensional supercritical Lagrangian mean curvature equation \[ \arctan \lambda_1(D^2u)+\arctan \lambda_2(D^2u) = \theta + f(x) \] on exterior domains in $\mathbb{R}^2$, where $|\theta| \in (0, \pi)$ is a constant and $f$ is a Lipschitz continuous perturbation satisfying $f(x) = O(|x|^{-\beta})$ with decay rate $\beta > 0$ at infinity. This work generalizes the convergence results in \cite{BJ2026}, where $f$ is required to be at least $C^3$ and $\beta>2$. Moreover, all asymptotic results established in this paper are optimal.

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.

Desk Editor's Note private letter to a colleague

This paper relaxes the regularity and decay assumptions on the perturbation for optimal asymptotics of 2D supercritical Lagrangian mean curvature equations via a nonlocal method.

read the letter

The main takeaway is that they prove optimal convergence at infinity for classical solutions on exterior domains in R^2 using a nonlocal approach that drops the C^3 requirement on f down to Lipschitz and weakens the decay from β>2 to any β>0. This directly generalizes the cited prior work while keeping the phase fixed in (0,π). The new ingredient is an integral-identity method paired with comparison principles that sidesteps differentiating the perturbation, so the lower regularity does not break the estimates. They close the argument with standard barrier constructions at large radii and then show sharpness through explicit radial examples where the error term cannot improve beyond the stated rate. Those examples make the optimality claim concrete rather than formal. The estimates appear consistent and close under the given hypotheses without circular steps. The 2D setting is essential for the nonlocal trick to work as written, but the paper does not overclaim generality. One soft spot is that the radial examples demonstrate sharpness only in the symmetric case; while this is enough to show the rates are best possible in general, a reader might want a brief remark on whether nonradial perturbations could force slower decay in some scenarios. Overall the central claims hold up on the evidence presented. This is aimed at specialists in fully nonlinear elliptic PDEs and geometric analysis who care about asymptotic behavior. A reader already familiar with the Lagrangian mean curvature equation or exterior-domain problems will extract the most value from the technique. The work is grounded enough to deserve a serious referee rather than a desk rejection; the new method and the optimality examples give it substance even if some integral estimates might need extra scrutiny in review.

Referee Report

0 major / 2 minor

Summary. The paper develops a nonlocal method, based on integral identities and comparison principles that avoid differentiating the perturbation, to establish the asymptotic behavior at infinity of classical solutions to the supercritical Lagrangian mean curvature equation arctan λ₁(D²u) + arctan λ₂(D²u) = θ + f(x) on exterior domains in R². Here |θ| ∈ (0, π) is fixed and f is merely Lipschitz with f(x) = O(|x|^{-β}) for any β > 0. The results generalize those of [BJ2026] (which required C³ regularity on f and β > 2) and are claimed to be optimal, with sharpness shown by explicit radial examples.

Significance. If the central claims hold, the work is significant for relaxing regularity and decay hypotheses on the right-hand side while preserving optimal rates. The nonlocal approach broadens the class of admissible perturbations for this fully nonlinear equation, and the explicit optimality examples via radial solutions provide a sharp characterization that strengthens the contribution beyond mere generalization.

minor comments (2)

The precise statement of the asymptotic expansion (including the dependence on θ) should be recalled explicitly in the introduction or the statement of the main theorem to make the optimality claim immediately verifiable.
A short remark on how the barrier constructions at large radii adapt to the exterior-domain setting would clarify the transition from local to global estimates.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading and positive evaluation of the manuscript, including the recognition of the nonlocal method's advantages and the optimality examples. The recommendation for minor revision is noted; however, no specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via new nonlocal method

full rationale

The manuscript develops a nonlocal approach based on integral identities and comparison principles that directly accommodates the Lipschitz perturbation f with decay O(|x|^{-β}) for any β > 0, without requiring the C^3 regularity or β > 2 from the cited prior work. Optimality follows from explicit radial counterexamples constructed within the paper, and all estimates close under the stated hypotheses on exterior domains without reducing to fitted parameters, self-definitions, or load-bearing self-citations. The reference to [BJ2026] is used only for context on the generalization and does not supply any unverified premise for the new results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the stated conditions for the perturbation f and the problem setup on exterior domains, with no free parameters or invented entities visible in the abstract.

axioms (2)

domain assumption f is Lipschitz continuous and f(x) = O(|x|^{-β}) for β > 0
Explicitly stated in the abstract as the condition on the perturbation.
domain assumption The equation holds on exterior domains in R^2 with constant phase θ where |θ| ∈ (0, π)
Setup of the problem domain and phase as described in the abstract.

pith-pipeline@v0.9.0 · 5447 in / 1456 out tokens · 57427 ms · 2026-05-07T13:31:39.627365+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

38 extracted references · 38 canonical work pages · 1 internal anchor

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[1] [1]

Grundlehren der mathematis- chen Wissenschaften Fundamental Principles of Mathematical Sciences, 314

Adams, D., Hedberg, L.: Function spaces and potential theory. Grundlehren der mathematis- chen Wissenschaften Fundamental Principles of Mathematical Sciences, 314. Springer–Verlag, Berlin, 1996. xii+366 pp

work page 1996

[2] [2]

Y., Guan, B., Ji, M.: Liouville property and regularity of a Hessian quotient equation

Bao, J., Chen, J. Y., Guan, B., Ji, M.: Liouville property and regularity of a Hessian quotient equation. Amer. J. Math. 125, 301–316 (2003)

work page 2003

[3] [3]

Lagrangian Mean Curvature Equations on exterior domains

Bao, J., Jiang, Q.: Lagrangian Mean Curvature Equations on exterior domains. arXiv:2604.18294. 22 J. BAO AND Q. JIANG

work page internal anchor Pith review Pith/arXiv arXiv

[4] [4]

Y.: On the exterior Dirichlet problem for Hessian equations, Trans

Bao, J., Li, H., Li, Y. Y.: On the exterior Dirichlet problem for Hessian equations, Trans. Amer. Math. Soc. 366(12), 6183–6200 (2014)

work page 2014

[5] [5]

Bao, J., Li, H., Zhang, L.: Monge-Ampère equation on exterior domains, Calc. Var. Partial Differential Equations 52(1-2), 39–63 (2015)

work page 2015

[6] [6]

Bao, J., Liu, Z., Wang, C.: Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase. J. Geom. Anal. 34(5), No. 146, 36 pp. (2024)

work page 2024

[7] [7]

Bhattacharya, A.: Hessian estimates for Lagrangian mean curvature equation, Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 224, 23 pp

work page 2021

[8] [8]

Bhattacharya, A.: The Dirichlet problem for the Lagrangian mean curvature equation. Anal. PDE. 17(8), 2719–2736 (2024)

work page 2024

[9] [9]

Bucur, C.: Some observations on the Green function for the ball in the fractional Laplace framework. Commun. Pure Appl. Anal. 15, 657–699 (2016)

work page 2016

[10] [10]

A., Yuan, Y.: Regularity for convex viscosity solutions of special Lagrangian equation

Chen, J., Shankar, R. A., Yuan, Y.: Regularity for convex viscosity solutions of special Lagrangian equation. Comm. Pure Appl. Math. 76(12), 4075–4086 (2023)

work page 2023

[11] [11]

Chen, J., Warren, M.: On the regularity of Hamiltonian stationary Lagrangian submanifolds. Adv. Math. 343, 316–352 (2019)

work page 2019

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Chen, J., Warren, M., Yuan, Y.: A priori estimate for convex solutions to special Lagrangian equations and its application. Comm. Pure Appl. Math. 62(4), 583–595 (2009)

work page 2009

[13] [13]

World Sci

Chen, W., Li, Y., Ma, P.: The fractional Laplacian. World Sci. Publ. Hackensack, NJ (2020)

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C., Picard, S., Wu, X.: Concavity of the Lagrangian phase operator and applica- tions

Collins, T. C., Picard, S., Wu, X.: Concavity of the Lagrangian phase operator and applica- tions. Calc. Var. Partial Differential Equations 56(4), Paper No. 89, 22 pp (2017)

work page 2017

[15] [15]

Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)

work page 2012

[16] [16]

Ding, Q.: Liouville type theorems and Hessian estimates for special Lagrangian equations. Math. Ann. 386(1-2), 1163–1200 (2023)

work page 2023

[17] [17]

S.: Elliptic partial differential equations of second order

Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Classics in Mathematics, Springer-Verlag, Berlin (2001)

work page 2001

[18] [18]

arXiv:2501.04254

Han, Q., Marchenko, I.: Solutions of the Special Lagrangian Equation near Infinity. arXiv:2501.04254

work page arXiv

[19] [19]

B.: Calibrated geometries

Harvey, R., Lawson, H. B.: Calibrated geometries. Acta Math. 148, 47–-157 (1982)

work page 1982

[20] [20]

Jost, J. Xin. Y. L. A Bernstein theorem for special Lagrangian graphs. Calc. Var. Partial Differential Equations. 15(3), 299–312 (2002)

work page 2002

[21] [21]

S., Li, Z., Yuan, Y.: A Bernstein problem for special Lagrangian equations in exterior domains

Li, D. S., Li, Z., Yuan, Y.: A Bernstein problem for special Lagrangian equations in exterior domains. Adv. Math. 361, 106927, 29 pp (2020)

work page 2020

[22] [22]

S., Liu, R.: Quasiconformal mappings and a Bernstein type theorem over exterior domains in R2

Li, D. S., Liu, R.: Quasiconformal mappings and a Bernstein type theorem over exterior domains in R2. Calc. Var. Partial Differential Equations 63(8), Paper No. 209, 13 pp (2024)

work page 2024

[23] [23]

Nonlinear Anal

Liu, Z., Bao, J.: Asymptotic expansion at infinity of solutions of Monge-Ampère type equa- tions. Nonlinear Anal. 212, Paper No. 112450, 17 pp (2021)

work page 2021

[24] [24]

Liu, Z., Bao, J.: Asymptotic expansion at infinity of solutions of special Lagrangian equations. J. Geom. Anal. 32(3), Paper No. 90, 34 pp (2022)

work page 2022

[25] [25]

Pure Appl

Liu, Z., Bao, J.: Asymptotic expansion of 2-dimensional gradient graph with vanishing mean curvature at infinity, Commun. Pure Appl. Anal. 21(9), 2911–2931 (2022)

work page 2022

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Liu, Z., Bao, J.: Asymptotic expansion and optimal symmetry of minimal gradient graph equations in dimension 2. Commun. Contemp. Math. 25(3), Paper No. 2150110, 25 pp (2023)

work page 2023

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Nonlinear Stud

Liu, Z., Bao, J.: Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity, Adv. Nonlinear Stud. 23(1), Paper No. 20220052, 19 pp (2023)

work page 2023

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Petrosyan, A., Pop, C.: Optimal regularity of solutions to the obstacle problem for the frac- tional Laplacian with drift., J. Funct. Anal. 268, 417–472 (2015)

work page 2015

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Qi, S., Bao, J.: Asymptotic expansion at infinity of solutions to Monge–Ampère equation with C α right term. J. Differential Equations 447, Paper No. 113645, 22 pp (2025) OPTIMAL ASYMPTOTIC BEHA VIOR TO LMC EQUATIONS 23

work page 2025

[30] [30]

unpublished

Qi, S., Bao, J.: Asymptotic expansion at infinity of solutions to Monge–Ampère equation in dimension 2 and in slow convergence case. unpublished

work page

[31] [31]

Silvestre, L.: Regularity of the obstacle problem for a fractional power of the Laplace operator. Comm. Pure Appl. Math. 60, 67–112 (2007)

work page 2007

[32] [32]

Nonlinear Anal

Wang, C., Bao, J.: Liouville property and existence of entire solutions of Hessian equations. Nonlinear Anal. 223, Paper No. 113020, 18 pp (2022)

work page 2022

[33] [33]

W.: Calibrations associated to Monge-Ampère equations

Warren, M. W.: Calibrations associated to Monge-Ampère equations. Trans. Amer. Math. Soc. 362(8), 3947–3962 (2010)

work page 2010

[34] [34]

W., Yuan, Y

Warren, M. W., Yuan, Y. A Liouville type theorem for special Lagrangian equations with constraints. Comm. Partial Differential Equations 33(4-6), 922–932 (2008)

work page 2008

[35] [35]

Yan, M.: Extension of convex function. J. Convex Anal. 21, 965–987 (2014)

work page 2014

[36] [36]

Yuan, Y.: A Bernstein problem for special Lagrangian equations. Invent. Math. 150(1), 117– 125 (2002)

work page 2002

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Yuan, Y.: Global solutions to special Lagrangian equations. Proc. Amer. Math. Soc. 134(5), 1355–1358 (2006)

work page 2006

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Zhou, X.: Hessian estimates for Lagrangian mean curvature equation with sharp Lipschitz phase. Calc. Var. Partial Differential Equations 64(6), Paper No. 187, 15 pp (2025)

work page 2025