0
$\begingroup$

Evans page 250&251

He gives us two theorems. Local approximation by smooth functions and global approximation by smooth functions. The second seems to give the first without any additional assumptions, what gives?

The global version says that $U$ is bounded, and the local does not say this. But immediately the local uses $U_\epsilon$ which is defined via the boundary of $U$ so evidently $U$ is bounded?

Then the only other difference seems to be that the approximating functions are $C^\infty$ on $U_\epsilon$ rather than on $U$. Why do we care about that? What do we gain by having these approximating functions defined on a greater region?

$\endgroup$
2
  • $\begingroup$ In the local theorem, $U$ is not necessarily bounded. For example, we can take $U=\mathbb R \times (-1,1)$ whose boundary is $(\mathbb R \times\{-1\})\cup(\mathbb R \times\{1\})$. In this case, $U_\varepsilon=\mathbb R\times(-1+\varepsilon, 1-\varepsilon)$ is unbounded too. $\endgroup$
    – Pedro
    Jul 10, 2017 at 10:08
  • $\begingroup$ @Pedro Oh right. Of course! $\endgroup$
    – F.White
    Jul 11, 2017 at 3:10

0

Reset to default

You must log in to answer this question.