Some doubts about Trace Theorem (for Sobolev functions).

Question

Let $\Omega\subset\mathbb{R}^N$ be a bounded regular domain and consider the Sobolec space $W^{1,p}(\Omega)$ for $p\in (1,\infty)$. I have some doubts with th trace theorem.

Roughly speaking, the Trace theorem states that there is a linear functional $$T:W^{1,p}(\Omega)\to L^p(\partial \Omega)$$

such that $Tu=u_{|\partial\Omega}$ for every $u\in C^\infty(\overline{\Omega})$.

My questions is:

The only functions for which $Tu=u_{|\partial\Omega}$ are the $C^\infty(\overline{\Omega})$ functions, or can we find, for instance, functions not continuous in $W^{1,p}(\Omega)$ such that the same equality holds?

Thank you

Answer 1

The reason that the equality $Tu=u_{|\partial \Omega}$ is stated for functions in $C(\overline{\Omega})$ is that for other functions it is not clear what $u_{|\partial \Omega}$ means. Of course, we can take any function $u\in W^{1,p}(\Omega)$ and extend its domain to $\overline{\Omega}$ by letting $u$ be equal to $Tu$ on the boundary. This will be an instance of $Tu=u_{|\partial \Omega}$ being true despite $u$ not necessarily being in $C(\overline{\Omega})$. But this is merely a tautology that does not teach us anything new.

But there are useful and nontrivial results along the lines of your question: starting with $\phi\in L^p(\partial \Omega)$, one can extend $\phi$ to a function in $\Omega$ (by solving the Dirichlet problem for some elliptic operator) and then recover $\phi$ from $u$ via nontangential limits. This does not require $\phi$ to be continuous. For a simple example, take $\Omega=\{z\in\mathbb C: |z|<1\}$ and $u=\log|z-1|$. Then $u\in W^{1,p}(\Omega)$ for $p<2$. The trace operator agrees with the boundary values of $u$ understood in the sense of nontangential limits.