$U = B(0,1)$, the open unit ball in $\mathbb{R}^n$ and $$ u(x) = |x|^{\alpha}$$ for $x \neq 0 $. And $u(0) := 0$. For which values of $\alpha < 0,n,p$ does $u$ belong to $W^{1,p}(U)$? (Here $W^{k,p}(U)$ is the Sobolev space on $U$.)
I'm guessing the solution should be $\alpha -1 > -d/p$. Here's my attempt:
$$\int_{B_1} u\phi = \int_{B_1-B_\epsilon} u\phi + \int_{B_\epsilon}u\phi.$$
And how do we go from here? My plan was to use integral by parts on the first term on the RHS and let $\epsilon$ goes to zero, so the second term should disappear, but I don't think it works with the integral by parts. Thanks.