Lagrangian Mean Curvature Equations on exterior domains

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 \[ \sum_{i=1}^{n} \arctan λ_i(D^2u) = θ+ f(x) \] on exterior domains in ${\mathbb{R}}^n$, where the constant $|\theta |\in ((n-2)\pi /2,n\pi /2)$, $n\ge 2$, 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 ($f\equiv 0$) 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 $n\ge 3$, 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.

46 pages