Let $\Omega\subset \mathbb{R}^{2}$ be a bounded, convex, polygonal domain and $H=\{(u_{1},u_{2})\in H^{\epsilon-\frac{1}{2}}(\Omega)\times H^{\epsilon-\frac{1}{2}}(\Omega),0<\epsilon<\frac{1}{2}:(div(u_{1},u_{2})\in L^{2}(\Omega)\}$. Now we define a norm $||.||_{H}$ on $H$ by, \ \begin{equation} ||((u_{1},u_{2})||_{H}=||div(u_{1},u_{2})||_{L^{2}(\Omega)}+||u_{1}||_{H^{\epsilon-\frac{1}{2}}(\Omega)}+||u_{2}||_{H^{\epsilon-\frac{1}{2}}(\Omega)}. \end{equation} Can we say that $C^{\infty}(\bar{\Omega})\times C^{\infty}(\bar{\Omega})$ is dense in $H$ with respect to the norm $||.||_{H}$ ?
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$\begingroup$ If it can be shown that $H(div,\Omega)$ is dense in $H$, then also it is okay. $\endgroup$– s.chowdhuryCommented Sep 23, 2015 at 9:32
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