Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains

Jiguang Bao; Qinfeng Jiang

arxiv: 2604.26246 · v2 · pith:5GJCBPUAnew · submitted 2026-04-29 · 🧮 math.AP

Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains

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 study the asymptotic behavior at infinity of solutions to the supercritical Lagrangian mean curvature equation \[ \sum_{i=1}^n \arctan \lambda_i(D^2u)=\theta+f(x) \] on exterior domains in \(\mathbb R^n\), \(n\ge 2\), where \(|\theta|>((n-2)\pi)/2\). The perturbation \(f\) is assumed to be locally Lipschitz near infinity and to satisfy a decay condition with rate \(\beta>0\). The main new ingredient is a scale-dependent difference quotient argument, combined with a nonlocal potential method, which avoids differentiating \(f\) twice and yields quantitative Hessian convergence under only Lipschitz regularity. We establish optimal asymptotic expansions in all dimensions and for all decay rates \(\beta>0\), including the critical logarithmic cases. This improves previous results requiring higher regularity of \(f\) and faster decay in \cite{BJ2026}.

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

Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains

Core claim

What carries the argument

If this is right

Where Pith is reading between the lines

Load-bearing premise

What would settle it

discussion (0)