A density problem for Sobolev spaces on Gromov hyperbolic domains

TL;DR

This paper demonstrates that in Gromov hyperbolic domains, Sobolev spaces with bounded derivatives are dense in those with integrable derivatives, and smooth functions are dense under additional geometric conditions, advancing the understanding of function approximation in these domains.

Contribution

It establishes density results for Sobolev spaces on Gromov hyperbolic domains, including planar and finitely connected cases, with conditions for smooth function density.

Findings

Sobolev space $W^{1,\, ext{infinity}}( ext{domain})$ is dense in $W^{1,p}( ext{domain})$ for $1 \,\leq p < \infty$.

In Jordan or quasiconvex Gromov hyperbolic domains, smooth functions are dense in $W^{1,p}( ext{domain})$.

Results apply notably to finitely connected planar domains.

Abstract

We prove that for a bounded domain $\Omega\subset \mathbb R^n$ which is Gromov hyperbolic with respect to the quasihyperbolic metric, especially when $\Omega$ is a finitely connected planar domain, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover if $\Omega$ is also Jordan or quasiconvex, then $C^{\infty}(\mathbb R^n)$ is dense in $W^{1,\,p}(\Omega)$ for $1\le p<\infty$.