# Sobolev spaces in $\mathbb R^n$ with functions having support on a closed set

I am interested in $H^s$ Sobolev spaces in $\mathbb R^n$ which have functions with support in a given closed set $K$ , denoted by $H^s_K$. Here $K$ is the complement of some bounded open set $\Omega$. For instance, $\Omega$ may satisfy the uniform cone condition or have Lipschitz boundary. An example is the quotient isometric isomorphism $H^s(\Omega) = H^s/{H^s_{\mathbb R^n\setminus\Omega}}$ .

A simple argument shows that these spaces are Hilbert spaces.

But what about dense spaces of smooth functions? For instance, is $C^{\infty}_K = \{f \in C^{\infty}_0 : \operatorname{supp}f \subset K\}$ dense in $H^s_K$? I think continuous functions on closed sets may introduce significant complications but I have seen papers which study them.

Just references will do.

• I edited your post to improve formatting. There is probably a missing not in "I have seen papers", but this is up to you to correct. – user53153 Mar 7 '13 at 16:45
• We can have a look at the book of Griesvard, Section 1.3.2, for some results in this direction. epubs.siam.org/doi/book/10.1137/1.9781611972030 – Voliar Feb 17 '17 at 14:44