I am currently reading the paper "The Inverse Mean Curvature Flow and the Riemannian Penrose Inequality" written by Gerhard Huisken and Tom Ilmanen.

https://projecteuclid.org/euclid.jdg/1090349447

I am wondering what versions of comparison principle for viscosity solution was used in Lemma 3.4 to derive the sup norm of the solution of

$$\nabla \cdot\left(\dfrac{\nabla u}{\sqrt{|\nabla u|^2+\varepsilon^2}}\right)=\sqrt{|\nabla u|^2+\varepsilon^2}.$$

It can be supposed that $u$: $\Omega$ $\to$ $\mathbb{R}$ is smooth for $\Omega \subset \mathbb{R}^n $.

The authors had constructed a continuous subsolution by using distance function which is not smooth on cut locus. So I believe that comparison principle for viscosity solution was used. I have read the user's guide for viscosity solution, but the conditions of comparison principle there do not hold. So I am wondering what versions of comparison principle for viscosity solution was used. Thank you!