I'm reading Evans book on PDEs and I'm not sure of the meaning behind the Trace Theorem for Sobolev spaces and what it aims to accomplish. The theorem is as follows:
Assume a domain $U$ is bounded and that $\partial U$ is $C^1$. Then there exists a bounded linear operator
$$T: W^{1,p}(U) \to L^p(\partial U)$$ such that
i) $Tu = u|_{\partial U}$ if $u \in W^{1,p}(U) \cap C(\overline{U}) $
ii) $||Tu||_{L^p(\partial U)} \le C ||u||_{W^{1,p}(U)}$
for each $u \in W^{1,p}(U)$, with $C$ depending only on $p$ and $U$.
I get that if $u$ is continuous we have already have a definition of $u$ on the boundary and this leads to i).
I don't see what is meant by ii) though. I read it as 'for any $u \in W^{1, p}$, the size of $T$ acting on $u$ in $L^p$ over the boundary $\partial U$ will always be less than or equal to a constant times the size of $u$ in $W^{1, p}$ over the entire domain $U$'.
- How is this useful, we haven't even given an explicit definition of $T$, we have only made a statement about its size? We are relating norms over different domains ($\partial U$ and $U$), this seems like a meaningless thing to do..so why do we do it?
- Furthermore, what is the relevance of $||Tu||_{L^p(\partial U)}$ being less than or equal to a constant times $||u||_{W^{1,p}(U)}$? The constant could be arbitrarily large so I don't see the point of making this statement?
The point is, that the property (ii) is equivalent to the linear(!) map $T \colon W^{1,p}(U) \to L^p(\partial U)$ being continuous. This allows us to construct a unique $T$ from (i) by approximating $W^{1,p}(U)$-functions by continuous ones.