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arXiv · 2104.08953

On density of compactly supported smooth functions in fractional Sobolev spaces

Bartłomiej Dyda·Michał Kijaczko
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Abstract

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.

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Bartłomiej Dyda, Michał Kijaczko. 2021-04-18. On density of compactly supported smooth functions in fractional Sobolev spaces. https://doi.org/10.1007/s10231-021-01181-8

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