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arxiv: 2604.18294 · v1 · submitted 2026-04-20 · 🧮 math.AP

Lagrangian Mean Curvature Equations on exterior domains

Jiguang Bao , Qinfeng Jiang This is my paper

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classification 🧮 math.AP
keywords Lagrangian mean curvature equationexterior domainviscosity solutionquasiconformal mappingPerron's methodasymptotic behaviorDirichlet problemsupercritical phase
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The pith

For n at least 3, viscosity solutions to the Lagrangian mean curvature equation exist and are unique on exterior domains when the perturbation decays faster than the inverse square.

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

The paper develops an extended exterior quasiconformal mapping technique to control the behavior of solutions at infinity for the equation summing arctangents of the Hessian eigenvalues equals a constant phase plus a decaying perturbation. It applies Perron's method to construct solutions to the Dirichlet problem outside a bounded uniformly convex inner domain while prescribing the far-field asymptotics. The proofs cover the supercritical phase with nonzero perturbation and the subcritical phase with zero perturbation, extending earlier results on the homogeneous special Lagrangian case to weaker boundary regularity. A reader would care because this provides a systematic way to handle geometric fully nonlinear equations on unbounded regions of space.

Core claim

The authors introduce an extended exterior (K, K', α0)–quasiconformal mapping method to study the asymptotic behavior at infinity of solutions to the supercritical phase Lagrangian mean curvature equation sum arctan λ_i(D²u) = θ + f(x) on exterior domains in R^n. With the phase satisfying |θ| in ((n-2)π/2, nπ/2), n at least 2, and f decaying as O(|x|^{-β}) for β greater than 2, they solve the corresponding Dirichlet problem outside a bounded uniformly convex domain via Perron's method. For n at least 3 they establish existence and uniqueness of viscosity solutions both in the supercritical phase with f not identically zero and in the subcritical phase with f identically zero, generalizing a

What carries the argument

The extended exterior (K,K',α0)–quasiconformal mapping method, which tracks the asymptotic behavior at infinity for the fully nonlinear equation by controlling the gradient and Hessian decay.

Load-bearing premise

The perturbation f must decay strictly faster than the inverse square of distance at infinity, and the inner domain must be bounded and uniformly convex.

What would settle it

An explicit example or numerical approximation of two distinct viscosity solutions with identical inner boundary data and the same prescribed asymptotics at infinity, for n=3 and a small nonzero f satisfying the decay.

read the original abstract

We introduce an extended exterior $(K,K^{\prime},\alpha_0)$--quasiconformal mapping method to study the asymptotic behavior at infinity of solutions to the supercritical phase Lagrangian mean curvature equation \[ \sum_{i=1}^{n} \arctan \lambda_i(D^2u) = \theta + f(x) \] on exterior domains in $\mathbb{R}^n$, where the constant $|\theta|\in((n-2)\pi/2,n\pi/2)$, $n\geq 2$, and $f=O(|x|^{-\beta})$ is a perturbation term with the sharp decay condition $\beta>2$ at infinity. Our work generalizes the classical exterior Bernstein-type theorem for the special Lagrangian equation ($f\equiv0$) established by Li--Li--Yuan [Adv. Math. (2020)]. Via Perron's method, we solve the corresponding Dirichlet problem outside a bounded, uniformly convex domain, prescribing asymptotic behavior at infinity. For $n \geq 3$, we establish existence and uniqueness of viscosity solutions in both the supercritical phase case with $f \not\equiv 0$ and the subcritical phase case with $f \equiv 0$. This extends earlier work by Li [Trans. Amer. Math. Soc. (2019)] on the exterior Dirichlet problem for the special Lagrangian equation ($f \equiv 0$) under weaker regularity assumptions on the interior boundary and boundary data.

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

0 major / 3 minor

Summary. The manuscript introduces an extended exterior (K, K', α₀)-quasiconformal mapping method to control the asymptotic behavior at infinity for solutions of the Lagrangian mean curvature equation ∑ arctan λ_i(D²u) = θ + f(x) on exterior domains in R^n (n ≥ 2), where |θ| lies in ((n-2)π/2, nπ/2) and f = O(|x|^{-β}) with β > 2. Via Perron's method, the authors solve the exterior Dirichlet problem outside a bounded uniformly convex domain and establish existence and uniqueness of viscosity solutions for n ≥ 3, both in the supercritical case with f ≢ 0 and the subcritical case with f ≡ 0. The work generalizes the exterior Bernstein theorem of Li-Li-Yuan (2020) and extends Li's (2019) exterior Dirichlet result under weaker assumptions on the interior boundary and data.

Significance. If the viscosity comparison principles and asymptotic estimates hold, the result is a meaningful extension in the theory of fully nonlinear elliptic equations of geometric type. It supplies existence/uniqueness for perturbed special Lagrangian equations on unbounded domains while relaxing boundary regularity, which may facilitate further study of calibrated submanifolds or mean-curvature-type flows in exterior settings. The combination of Perron's method with a tailored quasiconformal mapping technique for sharp decay is a concrete technical contribution.

minor comments (3)
  1. [Introduction] §1 (Introduction): the precise definition of the extended exterior (K, K', α₀)-quasiconformal mapping is deferred; a short paragraph recalling the classical quasiconformal notion and the new exterior modifications would improve readability for readers unfamiliar with the 2020 Li-Li-Yuan framework.
  2. [Main Theorem] The statement of the main theorem (presumably Theorem 1.1 or 1.2) should explicitly list the precise Hölder or C^{1,1} regularity assumed on the interior boundary ∂Ω and on the boundary data g; the abstract claims “weaker regularity” but the comparison with Li (2019) would be sharper if the exact improvement is quantified.
  3. [Viscosity Solutions] In the viscosity-solution section, the definition of the test functions for the Lagrangian mean-curvature operator should be written out once (even if standard) to avoid any ambiguity when the phase θ is supercritical.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive assessment. The referee's summary accurately reflects the main results on existence and uniqueness of viscosity solutions for the Lagrangian mean curvature equation on exterior domains. We appreciate the recommendation for minor revision. Since no specific major comments were listed under the MAJOR COMMENTS section, we have no individual points to address.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes existence and uniqueness of viscosity solutions to the Lagrangian mean curvature equation on exterior domains via Perron's method combined with an extended quasiconformal mapping technique for asymptotics. It explicitly generalizes results from independent prior works by Li--Li--Yuan (2020) and Li (2019), which are cited as external foundations rather than self-referential. No steps reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations; the phase restrictions, decay conditions on f, and boundary assumptions are stated independently and used to control the comparison principle and asymptotic behavior without circular reduction. The derivation chain is self-contained against external benchmarks in the cited literature.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Insufficient information from the abstract alone to identify concrete free parameters, axioms, or invented entities used in the proofs.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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

    math.AP 2026-04 unverdicted novelty 6.0

    Solutions to the supercritical Lagrangian mean curvature equation in 2D exterior domains exhibit optimal asymptotic behavior at infinity under Lipschitz perturbations decaying at any positive rate.

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