Some doubts on the Trace Theorem

Question

The Trace Theorem in Evan's Book (1st edition) says that,

Assume $U$ is bounded and $\partial U$ is $C^1$. Then there exsits a bounded linear operator $T$, $$T:W^{1,p}(U)\rightarrow L^p(U)$$ such that,

(i) $Tu=u|_{\partial U}$ if $u\in W^{1,p}(U)\cap C(U)$,

and

(ii) $\|Tu\|_{L^p(\partial U)} \le C \| u \|_{W^{1,p}(U)}$.

for each $u\in W^{1,p}(U)$, with the constant $C$ depending only on $p$ and $U$.

I am interested in the uniqueness. Does there exist different operators $T_1$ and $T_2$ which satisfy the condition in the Trace Theorem ? This theorem does not answer this question.

If yes, then we cannot define $Tu$ as the trace of $u$ on $\partial U$.

Answer 1

The trace operator $T$ is characterized by two facts:

I - $T$ is a bounded linear operator from $W^{1,p}(\Omega)$ into $L^p(\partial\Omega)$.

II - $T_{|C^1(\overline{\Omega})}$ is equal to the restriction operator to the boundary, i.e. if $u\in C^1(\overline{\Omega})$ then $$Tu=u_{\partial\Omega}$$

Item II guarantees that $T$ restricted to $C^1(\overline{\Omega})$ is unique. The first item guarantess that it is possible to extend $T$ to $W^{1,p}(\Omega)$ (remember that if $\partial\Omega\in C^1$ then $C^1(\overline{\Omega})$ is dense in $W^{1,p}(\Omega)$) and this extension is unique.

Answer 2

Since $C^1(\bar U)$ is dense in $W^{1,p}$, and $T$ is continuous on $W^{1,p}$, it follows that $T$ is uniquely determined.