I'm reading the [user's guide to viscosity solutions][1]. 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?
Edit: I noticed that in 4.2 we took the sup and in section 6 it is the limsup. I think that doesn't change the core of the question: Let $\mathcal{F}$ be a countable family of subsolutions and replace sup with limsup. The question remains: is simultaneous limsupping and *-ing the same as limsupping followed by *-ing?
[1]: https://arxiv.org/pdf/math/9207212.pdf
Best Answer
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 $x=0$.