A density problem for Sobolev spaces on planar domains

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 $1\le p<\infty$.

12 pages with 1 figure

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