A density problem for Sobolev spaces on planar domains

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

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