Question about trace operator

Question

From the general trace theorem we know for instance that if $f\in W^{1-\frac{1}{p},p}(\partial\Omega)$, then there exists a function $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega}=f$. But is it also true if $\partial\Omega$ is infact only a subset of the boundary, i.e. if there exists $f\in W^{1,p}(\Omega)$ such that $f|_{\partial\Omega'}=f$, where $\partial\Omega'\subset\partial\Omega$ and $f\in W^{1-\frac{1}{p},p}(\partial\Omega') $?

Answer 1

The trace theorem you mentioned is an "if and only if" characterization, hence the question you asked is equivalent to:

is it true that $f \in W^{1-\frac 1p,p}(D')$ extends to a function $\bar{f} \in W^{1-\frac 1p,p}(D)$ (here $D' \subseteq D = \partial \Omega$).

As in the case of extension theorems for standard Sobolev spaces, the answer depends on $D$.