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I am considering the Sobolev space $W^{1,\infty}(\Omega)$ on a bounded Lipschitz domain $\Omega \subseteq \mathbb{R}^2$. I am wondering whether the trace theorem holds in this case with constant one (independent of the domain), i.e. \begin{align} \| u \|_{L^{\infty}(\partial \Omega)} \le \| u \|_{W^{1,\infty}(\Omega)}. \end{align} It would be sufficient if smooth functions are dense in $W^{1,\infty}(\Omega)$, but the Meyers-Serrin Theorem is only stated for $p<\infty$ (at least on the german wikipedia site), so I'm not sure if this is true.

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    $\begingroup$ Lipschitz functions are uniformly continuous, thus the trace is defined at every point and it is clearly bounded by the L infinity norm over $\Omega$ $\endgroup$ – Piero D'Ancona Feb 22 '14 at 21:43

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