Comparison principle for viscosity solution

Question

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!

Answer 1

The two conditions in the user's guide are (3.13) and (3.14). The latter holds for your equation, but not the former, which is based on the existence of a zeroth order term $\gamma u$ in the PDE. Assumption (3.13) is only used to perturb a sub (or super) solution into a strict sub (or super) solution, which is the essential part of the comparison principle proof and leads to the contradiction.

Without (3.13) you still have strict comparison. That is, if $F(Du,D^2u) + \epsilon \leq 0$ and $F(Dv,D^2v) \geq 0$ in $\Omega$ in the viscosity sense for some $\epsilon>0$ then $\max_{\bar{\Omega}} (u-v) = \max_{\partial \Omega} (u-v)$. This only requires degenerate ellipticity ($F(p,X) \geq F(p,Y)$ whenever $X\leq Y$). This is briefly explained in Section 5C in the user's guide. To get comparison without strictness, you try to find a small perturbation (in $L^\infty$) of a subsolution that converts it into a strict subsolution, and send $\epsilon\to 0$. This strictification step requires some additional structure assumption on the PDE.

There are many tricks for doing the strictification step, and the trick to use depends on the type of equation at hand. The assumption (3.13) is the simplest condition to impose, and allows you to add or subtract small constants from $u$ to make $u$ a strict sub or super solution. In other cases the equation may be homogeneous in $D^2u$, like Monge-Ampere $-\text{det}(D^2u)=f$, and if $f>0$ then you can strictify by multiplying $u$ by a constant slightly less or greater than $1.$

Your equation does not have an obvious strictification trick, so my guess is that the construction of the subsolution actually gives a strict subsolution, and so the authors are using strict comparison. Looking at the proof briefly, there are a lot of inequalities that are loose and may actually be strict.