First, I give my motivation to ask this question. The generalised Neumann trace can be defined as $$ {}_{H^{-1/2}(\partial\Omega)}\langle\frac{\partial u}{\partial{\mathbf{n}}},v\rangle_{H^{1/2}(\partial\Omega)} ={}_{H^{-1}(\Omega)}\langle\Delta u,v\rangle_{H^1(\Omega)}-\int_{\Omega}\nabla u\cdot\nabla v. $$ But this involves integral in the volume of $\Omega$ which is not really like a trace to me. In particular, if we substitute this definition to Green's representation formula, we obtain an identity of no use like $0=0.$

Then, I found a theorem in Girault-Raviart's book says $H(\mathrm{div},\Omega)$ always has normal trace in $H^{-1/2}(\partial\Omega)$ by smooth approximation. This is the first time I saw such smooth approximation result of **negative** order Sobolev spaces. I searched on the web and in books but I found only for $H^{-s}(\mathbb{R}^n)$ that $C_c^\infty(\mathbb{R}^n)$ is dense. I can not find similar results on Lipschitz domains. It would be good to give me a reference.

Second, I also saw a result in McLean's book. That says, for any integer negative order Sobolev space $W^{-m,p}(\Omega)\ni f$ there is a representation
$$
f=\sum_{|\alpha|\leq m}\partial^{\alpha}f_{\alpha} \mbox{ with }f_{\alpha}\in L^p{(\Omega)}.
$$
But he does not say about negative **real** order Sobolev spaces. I would like a reference on similar results on negative **real** order Sobolev spaces.