Hi,

in which sobolev spaces , , are the functions

a) with

b) with ?

So, do I have to find out in which spaces the functions above has a first weak derivative?

And if yes: How can I find it out?

Thanks in advance,

Alexander

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- Dec 3rd 2011, 06:52 AMAlexanderWSobolev spaces, weak derivative
Hi,

in which sobolev spaces , , are the functions

a) with

b) with ?

So, do I have to find out in which spaces the functions above has a first weak derivative?

And if yes: How can I find it out?

Thanks in advance,

Alexander - Dec 3rd 2011, 08:31 AMgirdavRe: Sobolev spaces, weak derivative
If the function is differentiable in the usual sense, then it's also weakly differentiable and these two derivatives are the same.

- Dec 3rd 2011, 11:27 AMJose27Re: Sobolev spaces, weak derivative
Yes, so as

**girdav**noted, you have to check for which do the derivatives lie in .

For the first one, notice that so that iff . Now so for all , and (this is easy to see: change and bound below by a harmonic series). So that iff .

The second one is easier: and so that doesn't belong to any Sobolev space for . - Dec 3rd 2011, 02:07 PMAlexanderWRe: Sobolev spaces, weak derivative
Thank you very much for your helpful answers!

Bye,

Alexander - Dec 3rd 2011, 02:10 PMAlexanderWRe: Sobolev spaces, weak derivative
.