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 $u\in C(\bar{U})\cap L^p(U)$
PDE Evans, 1st edition, Chapter 5, Problem 14
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1There is another answer at this website, the link is: http://math.stackexchange.com/questions/332599/pde-question-in-evans – 2013-05-06