I have question on problem $14$ of Evan's book page $309.$ I know how to proceed it
by simply using the definition of $W^{1,n}.$ All I need to show is that $\|u\|_{L^n}$ and $\|Du\|_{L^n}$ are bounded. However, I have a very creative and naïve approach.
The problem in Evan's book looks like this:
Verify that if $n > 1$, the unbounded function $u = \log \log(1 + \frac{1}{|x|})$ belongs to $W^{1,n}(U),$ for $U = B(0,1).$
My approach:
Since we know that $\log(\log(x))$ is $C^1$ on $[1.5,\infty)$, we also know the derivative is bounded. Heuristically, Chain Rule might work here as long as $1+\frac{1}{|x|}$ is in $W^{1,n}(U).$ Then we can conclude that the composition $\log(\log(1+\frac{1}{|x|}))$ is in $W^{1,n}(U).$
But how do we know that $1+\frac{1}{|x|}$ is in $W^{1,n}(U)?$ The function does not admit any derivative at origin. What was wrong with my approach?
We can see that a function $u(x) = |x|^{-\alpha}$ is in $W^{1,n}$ if and only if $\alpha < \frac{n-n}{n} = 0,$ which means $-\alpha>0.$ We also know weak derivative is linear, hence $1+\frac{1}{|x|}$ is not a $W^{1,n}$ function. It implies that our proof was wrong. We cannot use the Chain Rule here.