A density problem for Sobolev spaces on planar domains

TL;DR

This paper proves the density of certain Sobolev spaces on planar domains, showing that smooth functions are dense in Sobolev spaces under specific geometric conditions, which advances understanding of approximation in these function spaces.

Contribution

It establishes the density of $W^{1,\infty}(\,\Omega)$ in $W^{1,p}(\Omega)$ for bounded simply connected domains and shows smooth functions are dense in Sobolev spaces on Jordan domains.

Findings

Density of $W^{1,\infty}(\Omega)$ in $W^{1,p}(\Omega)$ for simply connected domains.

Smooth functions are dense in $W^{1,p}(\Omega)$ on Jordan domains.

Extension of approximation results to planar Sobolev spaces.

Abstract

We prove that for a bounded simply connected domain $\Omega\subset \mathbb R^2$, the Sobolev space $W^{1,\,\infty}(\Omega)$ is dense in $W^{1,\,p}(\Omega)$ for any $1\le p<\infty$. Moreover, we show that if $\Omega$ is Jordan, then $C^{\infty}(\mathbb R^2)$ is dense in $W^{1,\,p}(\Omega)$ for $1\le p<\infty$.