A viscosity solution of a nonlinear partial differential equation is a continuous
function whose differential inequalities are tested using twice continuously
differentiable local comparison functions
that touch it from above or below. A comparison function 
touches 
from above at 
if 
and 
in a neighborhood
of 
.
Touching from below uses the reversed inequality (Crandall et al. 1992, §2).
For a degenerate elliptic equation 
with 
nonincreasing in its last matrix
argument, a test touching from above requires 
, and a test touching from below requires
.
Both inequalities are required for a solution. The function
itself need not initially possess the derivatives
appearing in 
.
For the unnormalized infinity Laplacian, use 
.
Thus an upper test requires 
, and a lower test requires 
, including tests with zero gradient.
