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arXiv · 1605.08260

A density problem for Sobolev spaces on Gromov hyperbolic domains

Pekka Koskela·Tapio Rajala·Yi Ru-Ya Zhang
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Abstract

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

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Pekka Koskela, Tapio Rajala, Yi Ru-Ya Zhang. 2016-05-26. A density problem for Sobolev spaces on Gromov hyperbolic domains. https://arxiv.org/abs/1605.08260

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