# Two types of limits of viscosity solutions

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?

New contributor
Zhanfeng Lim is a new contributor to this site. Take care in asking for clarification, commenting, and answering. Check out our Code of Conduct.