# Density in fractional Sobolev space

Suppose $$s∈(0,1)$$, $$D$$ is an open set in $$\mathbb{R}^d$$. Define $$H^s=(1−\Delta)^{-s/2}L^2\left(\mathbb{R}^d\right),$$ $$H^s_D=\left\{f\in H^s:f=0 \mbox{ a.e. on } D^c\right\}.$$ Q: Is $$C^\infty_c(D)$$ dense in $$H^s_D$$(with norm $$\|\cdot\|_{H^s}$$) for any open set $$D$$?

Is there any element reference?

• Not "Dobolev space", but "Sobolev space", in your title. I corrected your typo... – paul garrett Jan 23 at 23:15

Assuming that the boundary of $$D$$ is regular, say $$D$$ is a bounded domain with Lipschitz boundary, the density is true. Equivalent definition of $$H^2$$ is as follows: $$H^s=W^{s,2}$$, where $$\Vert f\Vert_{W^{2,s}}=\Vert f\Vert_2+\left(\int_{\mathbb{R}^d}\int_{\mathbb{R}^d} \frac{|f(x)-f(y)|^2}{|x-y|^{n+2s}}\right)^{1/2}$$ For a proof, see Section 3 in [1]. Now it is relatively easy to prove that a function in $$W^{2,s}$$ that vanishes in the complement of $$D$$ can be approximated by $$C_0^\infty(D)$$. This is more or less contained in Corollary 5.5 in [1].