On density of compactly supported smooth functions in fractional Sobolev spaces

TL;DR

This paper investigates conditions for the density of smooth, compactly supported functions in fractional Sobolev spaces, linking it to boundary geometry and providing explicit descriptions of closures and a Hardy inequality variant.

Contribution

It establishes new criteria based on Assouad codimension for density in fractional Sobolev spaces and characterizes the closure of smooth functions under geometric assumptions.

Findings

Density depends on boundary Assouad codimension.

Explicit description of the closure of smooth functions.

Proves a fractional Hardy inequality variant.

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.