Suppose comparison principle [CP] holds for the viscosity solution, then the unique solution can be characterized by the pointwise infimum of the piecewise smooth continuous viscosity supersolutions.
To proceed, we first define the space of supersolution, for any open set
We extend a function of into by . If there is no confuse, we shall use in replace of . Also define the space of continuous piecsewise smooth supersolution by
Proof: First, one can show an arbitrary is a supersoluiton by the similar to the proof of Proposition 2.8 of [CC95]. Since is continuous, Lemma 4.5 of [CIL92] implies is again a supersolution.
Suppose is not the subsolution. Then, there exists and a parabola
that touches at from the above, satisfying
By continuity of , there exists such that
Then, we conclude
which leads to a contradiction to the definition (2) of .