Two types of limits of viscosity solutions

Question

I actually posted this on math.stackexchange but it wasn't getting responses even after a bounty. I thought maybe it is too specialized so I'll post it here. I'm currently reading the user's guide to viscosity solutions. In Lemma 4.2 we define $w(x) = \sup\{u(x):u\in\mathcal{F}\}$, where $\mathcal{F}$ is a family of subsolutions to a certain equation. Next we consider the upper-semicontinuous envelope $w^*(x)$ and the lemma claims that $w^*$ is a subsolution as well. So far so good.

Later in section 6, we encounter a similar construction as before. Given a sequence $u_n(x)$ of subsolutions to an equation, we define the 'limit' $\bar{U}(z)=\limsup_{j\to\infty}\{u_n(x):n\geq j, |z-x|\leq\frac{1}{j}\}$, that is, we take the limsup and * operation simultaneously, instead of limsupping followed by * as before. Lemma 6.1 then claims that $\bar{U}(z)$ is a subsolution as well.

My question is, what is the difference between these two constructions; are there examples of sequences of solutions of functions whose 'limits' in the lemma 4.2 sense and the lemma 6.1 sense are different?

Answer 1

This is my answer from StackExchange:

The two operations apply to different objects, so although they share similarities, they are not strictly equivalent in any way I'm aware of. In the Perron method, one has a family of subsolutions $\mathcal{F}$, and it is the pointwise supremum over this family. In the end, one shows that $w(x)$ is a viscosity solution of the equation of interest, and so $w\in \mathcal{F}$, and the supremum is thus attained (and there is no limit).

On the other hand, the limsup operation applies to a sequence of functions. The context here is usually some approximation scheme for the viscosity solution (e.g., vanishing viscosity or finite difference schemes) where there is a natural ordering to the family of functions (e.g., increasing grid resolution, decreasing viscosity parameter).

The two operations indeed share a lot of similarities. They are both based on utilizing the maximum principle to pass to limits within the viscosity solution framework.

EDIT: To answer your edited question, taking the limsup and * separately gives a different operation. Consider the sequence of functions $u_n(x) = 1_{(0,1/n)}(x)$. Then $\limsup_n u_n(x)=0$, but the combined limsup and * operation gives a value of $1$ at $x=0$.