A Bernstein problem for special Lagrangian equations in exterior domains

Asymptotic behavior at infinity of solutions of Lagrangian mean curvature equations

“…In this section, we give an rigorous proof of Theorems 1.2, 1.3 and 1.4 based on Theorem 2.1 in [26]. For reading simplicity, we repeat the statement of this theorem.…”

“…First, we introduce the following result on asymptotic behavior of classical solutions of special Lagrangian equation (1.1) with τ = π 2 . See Theorem 1.1 of [26]. Theorem 2.2.…”

“…Now that we have obtained the limit of Hessian at infinity and it converge with a Hölder decay speed, it is time to capture the linear and constant part of the solution. In order to do this, we follow the line of Li-Li-Yuan [26] and analyze the linearized equation of (1.9). By the characterise result of ellipticity and concavity structure of F (λ(D 2 u)) type equation in [6], equation (1.9) is uniformly elliptic and concave.…”

“…We prove the following results that provide an asymptotic behavior at infinity. The idea is to use Legendre transform and apply the strategies in [26].…”

Maximal hypersurfaces over exterior domains

“…The Dirichlet problem for bounded domain was studied by Bartnik-Simon [BS82] and the isolated singularity problem was studied by Ecker [Ec86]. The exterior problem is a "complimentary" one for elliptic equations; see for example [Be51] [Si87] for minimal hypersurfaces, [CL03] for Monge-Ampere equation, [LLY17] for special Lagrangian equation and [HZ18] for infinity harmonic functions, besides the classic works such as [GS56] for linear ones. We study the exterior problem for the maximal surface equation in this paper.…”