If the function is differentiable in the usual sense, then it's also weakly differentiable and these two derivatives are the same.
in which sobolev spaces , , are the functions
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,
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 .