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My question is related to the mollifier properties in the appendix of Evans' PDE.

In proving that $f^{\varepsilon}$ is smooth, he constructs the difference quotient, with the original integral over $U.$ He then goes to the effort of writing the exact same line but changing the region of integration to V where V is some open set compactly contained within U.

I am wondering why he does this.

I'm sure it has something to do with the next line when he mentions uniform convergence of the mollifier's difference quotient to the partial derivative on V. Obviously I am a little lost and don't have a clear understanding of everything that's going on here.

Also, after he concludes the partial derivative of $f^{\varepsilon}$ exists, he writes it as an integral over U again. If someone is able to clear up my misunderstandings it would be greatly appreciated!

beedge89
  • 1,944
  • Presumably, the difference quotient does not converge unifomly on $U$, but just on compact subsets of $U$. So he has to introduce this $V$. Then he applies the most elementary result about interchanging a limit and an integral (i.e. uniform convergence plus integration on a bounded set), and finally goes back to $U$ to fit with the definition. – Etienne Jan 13 '15 at 07:44
  • Thanks for this: a few questions: 1) are we guaranteed to not have convergence on U? Or is it just that U was any open set so need to change to a compactly contained subset? And this leads to the next questions - why are the integrals over U and V equal? And is it because they are equal that he can just interchange as desired? – beedge89 Jan 13 '15 at 11:01
  • related: http://math.stackexchange.com/q/501383/9464 –  May 17 '16 at 01:13

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