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