We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{ s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{ R}^{ d}$. The density property is closely related to the lower and upper Assouad codimension of the boundary of $\Omega$. We also describe explicitly the closure of $C_{ c}^{ \infty}(\Omega)$ in $W^{ s,p}(\Omega)$ under some mild assumptions about the geometry of $\Omega$. Finally, we prove a variant of a fractional order Hardy inequality.
Categories: math.AP