Phragmén-Lindelöf-type theorems for functions in Homogeneous De Giorgi Classes

We study Phragmén-Lindelöf-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 $r\to \mathrm{\infty }$. By proper counterexamples, we show that in general we cannot expect $\alpha$ to be $1$.