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

TL;DR

This paper analyzes the asymptotic behavior at infinity of solutions to a two-dimensional supercritical Lagrangian mean curvature equation, extending previous results to less regular perturbations with optimal decay conditions.

Contribution

It introduces a nonlocal method to study solutions with weaker regularity and decay assumptions, generalizing prior convergence results and establishing optimal asymptotic behavior.

Findings

Extended convergence results to solutions with Lipschitz perturbations.

Proved optimality of asymptotic behavior in the supercritical phase.

Generalized previous results requiring higher regularity and decay.

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.