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 \({1\leqq p < \infty}\). Moreover, we show that if \({\Omega}\) is Jordan, then \({C^{\infty}({\mathbb{R}^{2}})}\) is dense in \({W^{1,\,p}(\Omega)}\) for \({1\leqq p < \infty}\).
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Communicated by C. De Lellis
This work was supported by the Academy of Finland via the Centre of Excellence in Analysis and Dynamics Research (Grant no. 271983).
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Koskela, P., Zhang, Y.RY. A Density Problem for Sobolev Spaces on Planar Domains. Arch Rational Mech Anal 222, 1–14 (2016). https://doi.org/10.1007/s00205-016-0994-y
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DOI: https://doi.org/10.1007/s00205-016-0994-y