I'm trying to do problem 8 from section 5.10 on Evans' PDE book. 

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> Basically the problem asks if $U$ is a bounded open subset of $\mathbb R^n$ with $C^1$ boundary and $u \in L^p (U)$, show that there does not exist a bounded linear operator from $L^p(U)$ to $L^p (\partial U)$ such that $Tu = u |\_{\partial U } $ whenever $u \in C(\bar U) \cap L^p (U)$.
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So my attempt is since the Trace Theorem is for all $u \in W^{1,p } (U)$. I have to find a sequence of function $u$ which is in $L^p (U)$ but $Du $ is not in $L^p $ and trying to make their $L^p ( \partial U)$ value blows up. I tried the function $f(x) = 1/|x|^\alpha$ with $p\alpha < n < p (\alpha + 1)$ and $U = B(0,1)$, but this approach doesn't work. I think I need to find functions with the above properties ($u \in L^p$ but $Du \notin L^p $) and grows rapidly near the boundaries but I don't know these functions. Any help is appreciated!