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Hi I am looking at the definition of standard mollifier $\eta$ in Evans, enter image description here

and the $\eta$ from wiki

Have a very basic question, is the $\eta$ in Evans also compactly supported? i.e. $\eta\in C_c^{\infty}$?

Proxava
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math101
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    yes. The support is ${x:|x|\le 1}$ – Thomas Jul 19 '15 at 06:58
  • Why does Evans claim $\operatorname{supp}(\eta_\epsilon) \subset B(0, \epsilon)$? Shouldn't it be $\operatorname{supp}(\eta_\epsilon) \subset \overline{B(0, \epsilon)}$? – fpmoo Oct 29 '22 at 11:21

1 Answers1

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Just look at the definition: since $\eta(x)=0$ for all $x \in \mathbb{R}^N$ with $|x| \geq 1$, it follows that $\operatorname{supp}\eta \subset \overline{B(0,1)}$. Hence $\eta \in C^\infty_c(\mathbb{R}^N)$.

Siminore
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