Phragm\'en-Lindel\"of-type theorems for functions in Homogeneous De Giorgi Classes

TL;DR

This paper investigates growth properties of functions in homogeneous De Giorgi classes, establishing power-like bounds on their maximum modulus and demonstrating limitations on growth rates through counterexamples.

Contribution

It provides Phragmén-Lindelöf-type theorems for these functions, revealing their growth behavior and the constraints on the exponent in their maximum modulus.

Findings

Maximum modulus grows at most polynomially with exponent less than 1

Counterexamples show growth rate cannot generally reach linear

Theorems extend classical growth results to De Giorgi classes

Abstract

We study Phragm\'en-Lindel\"of-type theorems for functions $u$ in homogeneous De Giorgi classes, and we show that the maximum modulus $\mu_+(r)$ of $u$ has a power-like growth of order $\alpha\in(0,1)$ when $r\to\infty$. By proper counterexamples, we show that in general we cannot expect $\alpha$ to be $1$.