A general result is that for a lipschitz bounded domain $\Omega$ in $R^n$, for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega)$, $1<p<\infty$, there exists $u\in W^{1,p}(\Omega)$ such that $u|_{\partial\Omega}=u^*$. But is it also true that for for $u^*\in W^{1-\frac{1}{p},p}(\partial\Omega')$, $\partial\Omega'\subset\partial\Omega$, there exists a $u\in W^{1,p}(\Omega)$ such that $u|_{\partial\Omega'}=u^*$?
Let me explain how the question comes from. Actually, if one reads papers of mechanics area, it will be assumed that the quantities have boundary values. What I want to do is to separate the variable such that we can deal with variables which have partial zero boundary values. Trivially, if for instance the bdry values are constant, we can always make this valid. However, I want to figure out some general results which generalize functions with non-trivial values. And we can also assume regularity of the partial boundary if this is needed. Here the Sobolev space is defined via Necas' definition, i.e. there exists charts $\Delta_r,r=1,...,m$ such that the boundary function $f(x_r',a_r(x_r'))$ defined on the boundary is of class $W^{{1-\frac{1}{p}},p}$, where $x_r'\in R^{n-1}$ is the coordinates of the local chart.
