Reading some lectures on Hamilton-Jacobi PDE theory I found some terminology that I really don't understand.

Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose that $u_\epsilon:\Omega\to \mathbb{R}$ is a function for $\epsilon>0$. Consider a sequence of function $(u_\epsilon)$ and let $u:\Omega\to\mathbb{R}$.

What does $u_\epsilon$ converges uniformly to $u$ as $\epsilon\to 0^+$ means?

I mean, I know what uniformly convergence means for a sequence of functions $(f_n)$ as $n\to +\infty$, but I really don't understand this terminology.

Thanks a lot in advance.


Let $\{u_h\}_h$ be a family of functions indexed by $h\in S\subset (0,\infty)$. We say that $u_h\to u$ uniformly as $h\to 0^+$ if for a given $\varepsilon>0$, we can find $\delta>0$ such that $$ \sup_{x\in\Omega}|u_h(x)-u(x)| < \varepsilon $$ for all $h\in S\cap(0,\delta)$.

Remark: Using this definition, we can see that $u_n\to u$ uniformly in the usual sense iff $\hat u_h\to u$ uniformly as $h\to 0^+$, provided that we take our index set to be $S=\{1,\frac 12,\frac 13,\dots\}$ and $$ \hat u_{1/n} := u_n. $$

  • $\begingroup$ That's completely clear! Thank you so much! $\endgroup$ – eleguitar Jan 12 at 18:45

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