# Is $H^s(\Omega) \cap C_0(\bar\Omega) \subset H^s(\Omega)$ dense?

We work on a bounded smooth domain. Let $C_0(\bar\Omega)$ be the set of continuous functions with compact support and let $s \in (0,1)$.

Is $H^s(\Omega) \cap C_0(\bar\Omega) \subset H^s(\Omega)$ dense?

I cannot find any reference..

• ok i probably misunderstood what is meant by $C_0(\bar\Omega)$ and deleted my answer – supinf Aug 23 '17 at 12:08
• How is $H^s(\Omega)$ defined in this case? – felipeh Aug 29 '17 at 1:32
• @felipeh we do it through the double integral definition, see Sobolev–Slobodeckij spaces in the Wikipedia article en.wikipedia.org/wiki/… – PostName Aug 30 '17 at 14:18
• I see. Since you do not assume anything about the behavior on the boundary, what I can already say is that $C_0(\bar{\Omega})$ is not dense when $s>\frac{1}{2}$. This is related to the fact that functions with jump discontinuities fail to be in $H^{1/2}$ and also to the continuity of the trace operator $H^s(\Omega)\to L^2(\partial\Omega)$ when $s>\frac{1}{2}$. I am not sure what happens at $s=\frac{1}{2}$ and below for your question, though I will try to answer. – felipeh Aug 30 '17 at 14:30