# Inverse trace theorem for partial trace

## 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.

• This will be true as soon you have an extension operator $W^{1-1/p,p}(\partial\Omega') \to W^{1-1/p,p}(\partial\Omega)$. Do you assume more on $\partial\Omega'$ than just "arbitrary subset"? – Willie Wong May 1 '16 at 16:25
• @WillieWong Nope. But can u tell me in which references I can find relevant contents? I think $\partial\Omega'$ should some how regular, but I dont know how to define the regularity for a manifold, since Im mostly deal with subsets in $R^n$. I think the regularity should some how relate to the charts or not? – Peter May 1 '16 at 17:27
• @WillieWong what I mean "subsets" is that we can deal with the sets directly, for instance ths sobolev spaces, without using a definition of charts. And I think there should exist some similar extension results for manifolds like the ones for Sobolev spaces of usual domains. – Peter May 1 '16 at 17:30
• In terms of simply extension theorems for Sobolev spaces, the recent works of Fefferman, Israel, and Luli (in some combination) come to mind. If your $\partial\Omega'$ is sufficiently nice you can extend to $\partial\Omega$ and apply the standard results. One of the problems however is with the definition of $W^{1 - 1/p,p}(\partial\Omega')$. If $\partial\Omega'$ is a submanifold of $\partial\Omega$ and the Sobolev space is the intrinsic one, then by taking a tubular nbhd the extension is trivial. – Willie Wong May 1 '16 at 19:56
• What I don't understand is your notation: if $\partial\Omega'$ is an arbitrary subset, for example, a discrete subset of $\partial\Omega$, how do you intend to define $W^{1-1/p,p}(\partial\Omega')$? Similarly if $\partial\Omega$ is a submanifold of positive codimension? Your question may admit a good technical answer, but first you have to specify what you mean. – Willie Wong May 1 '16 at 19:58