trace question 14 pde Evans chapter 5

Question

Let $U$ be bounded, with a $C^1$ boundary. Show that a ''typical'' function $u\in L^p(U) (1\leq p < \infty)$ does not have a trace on $\partial U$. More precisely there does not exist a bounded linear operator $$T:L^p(U)\rightarrow L^p(\partial U) $$ such that $Tu=u|_{\partial U}$ whenever $u\in C(\bar{U})\bigcap L^p(U)$

Answer 1

Hint: Construct an $L^p$ bounded sequence of continuous functions that look "nice" in the interior of $U$, but grow fast near the boundary. In order to construct this sequence explicitly, you can use the $C^1$ boundary condition in order to pass to local coordinates.