I try to understand the notions of weak derivative and Sobolev space
I take this example:
$f(x)= |x|\quad $ for $ \quad x\in [-1, 1]$
The derivative in the sense of distributions is $(T_f)^{'}= T_{f^{'}}+\delta_{-1}f(-1)+\delta_{1}f(1),$ where $T_f$ is the distributions associated to $f$ and $\delta_{a}$ is the Dirac distributions.
Now I want to compute $$\int_{-1}^{1}(f^{'}(x))^{2} dx$$ where$ f^{'}$ is the weak derivative of $f$ . but I don't know from where to start because this for me doesn't make sense.
Thank you in advance!
On $(-1, 1)$ the weak derivative of $f(x) = |x|$ is $f'(x) = \operatorname{sign}(x),$ which equals $1$ if $x>0$ and equals $-1$ if $x<0.$
The square of this equals $1$ on all of $(-1, 1)$ so the integral has value $2$.