Weak derivative (Sobolev spaces)

Question

I'm reading "Functional Analysis" - Michel Willem and I can't understand the definition of weak derivative from chapter 6, namely the definition of $$ \partial^\alpha f. $$ Can you give me a concrete example?

Thank you!

Answer 1

Let $f(x)=|x|$, $g(x)=-1$ if $x<0$, $g(x)=1$ if $x\ge0$. For any differentiable function $\phi\colon\mathbb{R}\to\mathbb{R}$ with compact support contained on an interval $[a,b]$ we have $$ \int_{-\infty}^\infty f(x)\,\phi'(x)\,dx=-\int_{\infty}^0x\,\phi'(x)\,dx+\int_0^{\infty}x\,\phi'(x)\,dx=\int_{\infty}^0\phi(x)\,dx-\int_0^{\infty}\phi(x)\,dx, $$ where we have used integration by parts and the fact that $\phi$ has compact sport, and hence $\phi(\pm\infty)=0$.

On the other hand $$ \int_{-\infty}^\infty g(x)\,\phi(x)\,dx=-\int_{\infty}^0\phi(x)\,dx+\int_0^{\infty}\phi(x)\,dx. $$ We have proved that $$ \int_{-\infty}^\infty f(x)\,\phi'(x)\,dx=-\int_{-\infty}^\infty g(x)\,\phi(x)\,dx $$ and this means that $f'=g$ in the sense of distributions.