Density problem for Sobolev spaces on Gehring Hayman domains with the ball separation condition in metric measure spaces

TL;DR

This paper proves that in certain metric measure spaces satisfying specific geometric conditions, the space of essentially bounded Sobolev functions is dense in the standard Sobolev space, facilitating approximation and analysis.

Contribution

It establishes the density of $N^{1, ext{infty}}( ext{Omega})$ in $N^{1,p}( ext{Omega})$ under Gehring Hayman and ball separation conditions in PI spaces, a new result in this setting.

Findings

Density of $N^{1, ext{infty}}( extOmega)$ in $N^{1,p}( extOmega)$ for $1 < p < extinfty$

Validation of geometric conditions for Sobolev space approximation

Extension of classical results to metric measure spaces with specific domain conditions

Abstract

We prove that for a domain $\Omega$ in a PI space $X$ such that $\Omega$ satisfies the Gehring Hayman condition and the ball separation condition, the Newtonian Sobolev space $N^{1,\infty}(\Omega)$ is dense in the space $N^{1,p}(\Omega)$ for $1 < p < \infty$.