quasiconformal mappings and a Bernstein type theorem over exterior domains in $\mathbb{R}^2$

TL;DR

This paper studies the behavior of quasiconformal mappings in exterior domains in the plane, establishing estimates and asymptotics, and applies these results to prove a Bernstein type theorem for certain nonlinear elliptic equations.

Contribution

It provides new Hölder estimates and asymptotic analysis for quasiconformal mappings, leading to a novel Bernstein type theorem for fully nonlinear elliptic equations in two dimensions.

Findings

Hölder estimates for quasiconformal mappings in exterior domains

Asymptotic behavior characterization at infinity

Bernstein type theorem for nonlinear elliptic equations

Abstract

We establish the H\"{o}lder estimate and the asymptotic behavior at infinity for $K$-quasiconformal mappings over exterior domains in $\mathbb{R}^2$. As a consequence, we prove an exterior Bernstein type theorem for fully nonlinear uniformly elliptic equations of second order in $\mathbb{R}^2$.