  We establish two Phragm\'en--Lindel\"{o}f theorems for a fully nonlinear
elliptic equation. We consider a dynamic boundary condition that includes both
spatial variable and time derivative terms. As a spatial term, we consider a
non-linear Neumann-type operator with a strict monotonicity in the normal
direction of the boundary on the spatial derivative term. Our first result is
for an elliptic equation on an epigraph in $\mathbb{R}^n$. Because we assume a
good structural condition, which includes wide classes of elliptic equations as
well as uniformly elliptic equations, we can benefit from the strong maximum
principle. The second result is for an equation that is strictly elliptic in
one direction. Because the strong maximum principle need not necessarily hold
for such equations, we adopt the strategy often used to prove the weak maximum
principle. Considering such equations on a slab we can approximate the
viscosity subsolutions by functions that strictly satisfy the viscosity
inequality, and then obtain a contradiction.

Analysis of PDEs