Examples where $1 \in W_0^{k,p}\left( U \right)$

Question

$M$ is a Riemannian manifold, $U$ is a domain in $M$. Consider the Sobolev space $W_0^{k,p}\left( U \right)$: the closure of $C_0^\infty \left( U \right)$ (smooth functions with compact support) in ${W^{k,p}}\left( U \right)$. Then are there examples where $1 \in W_0^{k,p}\left( U \right)$?

Answer 1

Let $M$ be closed and compact. Let $U$ be a connected component of $M$. It is open and connected, and hence is a domain (by most definitions). Clearly $1 \in W^{k,p}_0(U)$.

Answer 2

If a function belongs to $W^{k,p}_0(U)$, then its extension by $0$ outside of $U$ belongs to $W^{k,p}(M)$. (Proof here). Is it possible for $\chi_{U}$ to be in $W^{k,p}(M)$? Certainly not if $U$ has a nonempty smooth boundary, since $\chi_{U}$ would have a jump discontinuity across that boundary, and therefore would not have an ACL representative. On the other hand, if $K$ is a compact set of zero $W^{k,p}$ capacity, then $1\in W^{k,p}_0(M\setminus K)$. This is pretty much by the definition of capacity: we can find a function with small $W^{k,p}$ norm which is $1$ in a neighborhood of $K$, and subtract it from $1$ to get an approximation to $\chi_{M\setminus K}$.

Simple example: $U$ is the complement of a finite subset of $\mathbb R^n$ (or $n$-sphere if you prefer), and $kp<n$. Indeed, under assumption $kp<n$ the function $|x|^{-\epsilon}$ belongs to $W^{k,p}$ for small enough $\epsilon>0$. Truncating and scaling as appropriate, we find that the capacity of a point is zero.

Same works for $kp=n\ge 2$, but one has to use something like $\log^{1/2} |x|^{-1}$ in place of $|x|^{-\epsilon}$