I am currently reading Chapter 2 from the book Finite Elements by Dietrich Braess. Note that Sobolev Spaces are quite a new concept for me. I am trying to figure out how to approach the following problem (1.11 in the book):
Let $\Omega\subset\mathbb{R}^n$ be a sphere with the center at the origin. Show that $u(x)=\|x\|^s$ possesses a weak derivative in $L_2(\Omega)$ if $2s>2-n$ or if $s=0$.
I know that on $\mathbb{R}^n\setminus\{0\}$, the function possesses strong first order partial derivatives: $$\partial x_i(\|x\|^s)=s\|x\|^{s-2}x_i.$$ How can I prove that these verify the weak derivative condition: $$\int_{\Omega}\phi(x)s\|x\|^{s-2}x_i\,dx=-\int_\Omega\partial x_i(\phi)\|x\|^s\, dx$$ for any $\phi\in C_0^\infty(\infty)$? I think the coarea formula may be involved given that the domain is a sphere and we have the norm terms.
Also, can a function possess weak derivatives of higher order but not of first order? Or can I state that a function possesses or not weak derivatives only by looking at first order weak derivatives.