Asymptotic behavior at infinity of solutions of Lagrangian mean curvature equations

TL;DR

This paper investigates the asymptotic behavior of solutions to Lagrangian mean curvature equations with quadratic growth in exterior domains, extending classical Liouville theorems and relaxing growth conditions when the right-hand side is constant at infinity.

Contribution

It provides new asymptotic analysis results for Lagrangian mean curvature equations, including cases with constant asymptotic behavior, extending classical theorems in the field.

Findings

Established asymptotic behavior of solutions with quadratic growth.

Extended Liouville-type theorems for exterior domains.

Relaxed growth conditions when the source term is constant at infinity.

Abstract

We studied the asymptotic behavior of solutions with quadratic growth condition of a class of Lagrangian mean curvature equations $F_{\tau}(\lambda(D^2u))=f(x)$ in exterior domain, where $f$ satisfies a given asymptotic behavior at infinity. When f(x) is a constant near infinity, it is not necessary to demand the quadratic growth condition anymore. These results are a kind of exterior Liouville theorem, and can also be regarded as an extension of theorems of Pogorelov, Flanders and Yuan.