Infinity harmonic functions over exterior domains

TL;DR

This paper investigates infinity harmonic functions in exterior domains, demonstrating their asymptotic behavior and establishing conditions for solving Dirichlet problems in such settings.

Contribution

It provides new insights into the asymptotic nature of infinity harmonic functions and proves solvability of Dirichlet problems in exterior domains.

Findings

Infinity harmonic functions are asymptotic to planes or cones at infinity.

Solvability of Dirichlet problems is established for exterior domains.

Functions with linear growth at infinity are characterized.

Abstract

In this paper, we study the infinity harmonic functions with linear growth rate at infinity defined on exterior domains. We show that such functions must be asymptotic to planes or cones at infinity. We also establish the solvability of Dirichlet problems for exterior domains.