Existence of entire solutions to the Lagrangian mean curvature equations in supercritical phase

TL;DR

This paper proves the existence and uniqueness of entire solutions to Lagrangian mean curvature equations with supercritical phase functions, establishing uniform estimates and optimal convergence rates.

Contribution

It introduces new methods for constructing barriers and proves the existence, uniqueness, and optimal convergence rates for solutions in supercritical phases.

Findings

Existence and uniqueness of solutions under supercritical phase conditions

Construction of barrier functions for uniform estimates

Optimal convergence rate of phase functions

Abstract

In this paper, we establish the existence and uniqueness theorem of entire solutions to the Lagrangian mean curvature equations with prescribed asymptotic behavior at infinity. The phase functions are assumed to be supercritical and converge to a constant in a certain rate at infinity. The basic idea is to establish uniform estimates for the approximating problems defined on bounded domains and the main ingredient is to construct appropriate subsolutions and supersolutions as barrier functions. We also prove a nonexistence result to show the convergence rate of the phase functions is optimal.