Why is the ellipticity condition important in the theory of viscosity solutions? A fundamental assumption in User's guide (p. 2) is that the operator $F$ should be proper. 
However, the role of this monotonicity condition (and especially of the "ellipticity condition" part) is not quite clear to me. Why is it needed and why is it important?
 A: $\DeclareMathOperator{\Sym}{Symm}$ $\newcommand{\bR}{\mathbb{R}}$ Denote by $\Sym(n)$ the space of symmetric $n\times n$ real matrices equipped with the usual partial order. You should think of weakly increasing functions
$$f : \Sym(n) \to\bR. $$
Pick such a function $f$ and consider the  differential operator  $L_f$ $\DeclareMathOperator{\Hess}{Hess}$ that associates to a real valued function $u$ the  function
$$ L_f(u):= f\Big( \Hess(u)\Big). $$
For  example, $\DeclareMathOperator{\tr}{tr}$ if $f(A)=\tr A$  then
$$L_f(u)=\Delta u. $$
In general if $f$ is weakly increasing,  then $L_f$ will be degenerate elliptic. (You have to check this. I'm running out of time.)
A: A couple of points come to mind:


*

*Existence and uniqueness requires something like invertibility of the operator. In the case of something like
$$
F(x, u, \nabla u, \nabla^2 u) = 0
$$
the linearisation (about $u_0$ in direction $V$):
$$
A(x, V, \nabla V, \nabla^2 V) = \partial_t|_{t=0} \> F(x, u_0 + t V, \nabla(u_0 + tV), \nabla^2 (u_0 + tV))
$$
is a linear operator for $V$ and the monotonicity should correspond to this linear operator being elliptic.
In other words, the coefficient of $\nabla^2 V$ is exactly $\partial_z F$ where $F = F(x, y, p, z).$ Monoticity in the $z$ variable is the same as $\partial_z F \geq 0$ is the same as ellipticity of the linearised equation
Roughly speaking one might then hope to obtain existence and uniqueness for the original equation by first solving the linear equation (use the ellipticity!) and then employing the Inverse Function Theorem. 
Note: I doubt existence/uniqueness is obtained exactly this way in the guide, but the linearisation idea serves as good motivation for why you would want/need monotonicity. Notice that for linear equations, monotonicity is exactly ellipticity.

*The comparison principle (which is the generalisation of the maximum principle for fully non-linear equations) relies on exactly this monotonicity. Check the proof in this guide and you will see. Uniqueness of solutions does follow from the comparison principle (this is how the guide does it if I remember) in a similar way to how one may prove uniqueness for linear equations using the (strong) maximum principle.
