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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 $\partial U$.

Michael
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  • By stating the question in this way, you're pretty much limiting the possible responders to those who have that particular book handy. In fact you didn't even bother to include the name of the book, so some might not even realize they have the book you mean. If you took the trouble to state the actual theorem, you'd be more likely to get an answer. – Robert Israel Dec 16 '13 at 11:45
  • Sorry about that. I have edited it. Thank you！ – Michael Dec 16 '13 at 12:35

2 Answers2

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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.

mrf
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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.

Tomás
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