Lagrangian Mean Curvature Equations on exterior domains

TL;DR

This paper studies the asymptotic behavior and solves the exterior Dirichlet problem for supercritical phase Lagrangian mean curvature equations, extending classical results and establishing existence and uniqueness of solutions with decay conditions.

Contribution

It introduces an extended quasiconformal mapping method and generalizes the exterior Bernstein theorem for the Lagrangian mean curvature equation, including cases with perturbations.

Findings

Established asymptotic behavior of solutions at infinity.

Proved existence and uniqueness of viscosity solutions for supercritical and subcritical phases.

Extended earlier work to weaker boundary regularity conditions.

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.