Question on the definition of trace operator

Question

Let $\Omega$ be a domain with $C^1$-boundary and $1 \leq p < \infty$. Then there's exactly one bounded linear operator $$tr_{ \partial \Omega}: W^1_p(\Omega) \rightarrow L^p(\partial \Omega), ~ u \mapsto u|_{\partial \Omega}$$

for all $u \in C^1_b(\bar \Omega) \cap W^1_p(\Omega)$.

Our lecturer said that $\partial \Omega$ is a set of measure zero and as such, this map would not be welldefined if we only had $u \in W^1_p(\Omega)$, but with $u \in C^1_b(\bar \Omega)$ as well, it is well-defined.

Could someone please explain this statement to me?

Answer 1

I believe what your lecturer meant is that functions in $W^1_p$ are only defined up to sets of measure zero. That's because functions in $W^1_p$ are technically equivalence classes of functions. Thus, you can't simply define the trace operator to be the value of the function on the boundary, because it has measure zero.

This definition however works for functions that are also continuous. The trick then is to show that the trace operator is continuous, and to extend the definition of the operator to all of $W^1_p$ using the fact that $C^1$ is dense in $W^1_p$.

Here's how the extension of the operator to $W^1_p$ works. Suppose we have already proved the following inequality for functions $u\in C^1(\Omega)$: $$ \|\left. u\right|_{\partial\Omega}\|_{L^p(\partial\Omega} \leq \|u\|_{W^1_p(\Omega)}. $$ Now suppose we want to define the trace $\left.v\right|_{\partial\Omega}$ for some $v\in W^1_p$ which is not necessarily in $C^1(\Omega)$. What we can do is find a sequence of functions $u_n$ which converges to $v$ in $W^1_p$ (since continuous functions are dense in $W^1_p$). Then we have $$ \|\left. (u_n-u_m)\right|_{\partial\Omega}\|_{L^p(\partial\Omega} \leq \|u_n-u_m\|_{W^1_p(\Omega)}. $$ Thus the sequence of the traces on $\partial\Omega$ is a Cauchy sequence in $L^p(\partial\Omega)$. We can take its limit to be the definition of the trace for $v$. This definition now allows us to consider the trace operator as a continuous linear operator defined on all of $W^1_p$.