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I am trying to understand the proof of the following statement that is presented in the book “Finite Element Methods for Maxwell's Equations” by Peter Monk. The original source of the proof is a 1997 paper Un résultat de densité pour les équations de Maxwell by Ben Belgacem, Bernardi, Costabel, Dauge. Unfortunately, in this paper, written in French, it is even less explained why the steps in question are valid.

First we define the following sets ($\nu$ denotes the normal vector on the boundary): \begin{align*} \newcommand{\curl}{\operatorname{curl}} H_{\text{imp}}(\curl,\Omega) &= \{u \in H(\curl,\Omega) \;|\; u \times \nu \in L^2_t(\partial \Omega)\}, \\ \hat{H}_{\text{imp}}(\curl,\Omega) &= \{u \in H_{\text{imp}}(\curl,\Omega) \;|\; \curl u \cdot \nu = 0\}, \end{align*} where $L^2_t(\partial\Omega)$ denotes all tangential $L^2$ functions on the boundary (i.e. $u\cdot \nu = 0$).

Lemma 3.53. The space $\mathcal{C}^{\infty}(\bar{\Omega})$ is dense in $\hat{H}_{\text{imp}}(\curl,\Omega)$ (w.r.t. $\lVert u\rVert_{H_{\text{imp}}}^2 = \lVert u\rVert_{L^2}^2 + \lVert\curl u\rVert_{L^2}^2 + \lVert u \times \nu\rVert_{L^2(\partial\Omega)}^2 $).

The statement itself is already a little bit imprecise, because I think it is not meant that $\mathcal{C}^{\infty}(\bar{\Omega}) \cap \hat{H}_{\text{imp}}(\curl,\Omega)$ is dense in $\hat{H}_{\text{imp}}(\curl,\Omega)$, but rather that for every $f \in\hat{H}_{\text{imp}}(\curl,\Omega)$ there exists a sequence $(f_n)_{n\in\mathbb{N}}$ in $\mathcal{C}^{\infty}(\bar{\Omega})$ that converges to $f$ w.r.t. $\lVert\cdot\rVert_{H_{\text{imp}}}$. ($f_n$ does not need to satisfy $\curl f_n \cdot \nu = 0$.)

If it helps I can also copy the proof of the statement, but I think I can isolate my questions.

There are two steps that are (at least for my understanding) not sufficiently explained:

First step

If $(\nu \times \nabla p) \times \nu \in L^2(\partial\Omega)$ (the tangential trace of $\nabla p$ as an element of $H(\curl,\Omega)$), then $\nabla_{\partial\Omega} p$ exists ($p\bigr\rvert_{\partial\Omega} \in H^1(\partial \Omega)$) and $\nabla_{\partial\Omega} p = (\nu\times \nabla p) \times \nu$.

I can see that the reverse is true: If $\nabla_{\partial\Omega}p$ exists, then $\nabla_{\partial\Omega} p = (\nu\times \nabla p) \times \nu$. However, I don't see how one can show the claimed direction. Maybe, if you can provide me some good references of $H^1$ spaces on Lipschitz boundaries I can come further.

Second step

For notational simplicity I denote that tangential trace by $\pi_{\tau}$ ($\pi_{\tau} f = (\nu \times f) \times \nu$). The second step that I don't understand is:

If $$ \langle A \times \nu, \pi_{\tau} \phi\rangle = \langle u \times\nu, \pi_{\tau} \phi \rangle \quad\text{for all}\quad \phi \in X_{T} = H(\curl,\Omega) \cap H_0(\operatorname{div},\Omega), $$ then $A \times\nu = u \times \nu$.

I am aware that the normal component, that is zero for all $\phi \in X_T$, is not relevant. However, this would still mean that $\pi_{\tau} X_T$ is dense in $\pi_\tau H(\curl,\Omega)$, which is not immediately clear to me. Maybe $X_T$ is even dense in $H(\curl,\Omega)$, but I don't see that.

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