Optimal Asymptotic Behavior at Infinity of Solutions to the Lagrangian Mean Curvature Equation with Supercritical Phase in Dimension Two
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866909000018165760 |
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| author | Bao, Jiguang Jiang, Qinfeng |
| author_facet | Bao, Jiguang Jiang, Qinfeng |
| contents | We employ a nonlocal method to study the asymptotic behavior at infinity ofsolutions to the two-dimensional supercritical Lagrangian mean curvature equation
\[
\arctan λ_1(D^2u)+\arctan λ_2(D^2u) = θ+ f(x)
\] on exterior domains in $\mathbb{R}^2$, where $|θ| \in (0, π)$ is a constant and $f$ is a Lipschitz continuous perturbation satisfying $f(x) = O(|x|^{-β})$ with decay rate $β> 0$ at infinity. This work generalizes the convergence results in \cite{BJ2026}, where $f$ is required to be at least $C^3$ and $β>2$. Moreover, all asymptotic results established in this paper are optimal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_26246 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Optimal Asymptotic Behavior at Infinity of Solutions to the Lagrangian Mean Curvature Equation with Supercritical Phase in Dimension Two Bao, Jiguang Jiang, Qinfeng Analysis of PDEs 35J60, 35C20 We employ a nonlocal method to study the asymptotic behavior at infinity ofsolutions to the two-dimensional supercritical Lagrangian mean curvature equation \[ \arctan λ_1(D^2u)+\arctan λ_2(D^2u) = θ+ f(x) \] on exterior domains in $\mathbb{R}^2$, where $|θ| \in (0, π)$ is a constant and $f$ is a Lipschitz continuous perturbation satisfying $f(x) = O(|x|^{-β})$ with decay rate $β> 0$ at infinity. This work generalizes the convergence results in \cite{BJ2026}, where $f$ is required to be at least $C^3$ and $β>2$. Moreover, all asymptotic results established in this paper are optimal. |
| title | Optimal Asymptotic Behavior at Infinity of Solutions to the Lagrangian Mean Curvature Equation with Supercritical Phase in Dimension Two |
| topic | Analysis of PDEs 35J60, 35C20 |
| url | https://arxiv.org/abs/2604.26246 |