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it is well known, that for a non-convex domain $\Omega$ the space $H^1(\Omega, \mathbb{R}²)$ is a proper subset of $H(curl) \cap H(div)$. Here, $H(curl) = \{v \in L²(\Omega)², \nabla\times v = \partial_1v_2-\partial_2v_1 \in L²(\Omega)\} $ and $H(div)=\{v \in L²(\Omega)², \text{div } v = \partial_1v_1+\partial_2v_2 \in L²(\Omega)\}$.

I am looking for an example, where $ v \in H(curl) \cap H(div)$, but not in $H^1(\Omega, \mathbb{R}²)$. Furthermore, I want to have the constraint div v=0. Lets consider $\Omega=(-0.5,0.5)²\backslash [0,0.5]²$.

In Paper, there might be an example given by equation (5.2) $v=\nabla \times (r^\frac{2}{3} \text{ cos}(\frac{2}{3} \theta-\frac{\pi}{3}))$. Is this one an example, I am looking for?

Thank you very much for your help.

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The example you give from the paper is a little bit obscure to me. I propose another solution/perspective, which is quite general (but does not provide an explicit expression, at least in principle) for what concerns counter-examples in the context of $H(\text{curl}),H(\text{div})$. The main idea is to go back to the Laplace equation, which is very easy to deal with, and everything is known about it.

Consider the (weak) solution $u \in H^1(\Omega)$ of the boundary value problem

$$ \begin{cases} \Delta u = 0 \qquad \text{in } \Omega \\ u = g \qquad \text{on } \ \partial \Omega, \end{cases} $$ where $g \in H^{1/2}(\partial \Omega)$. It is well-known that $u \notin H^2(\Omega)$ if the domain is not enough regular (non-convex, with non-smooth boundary...). Define now

$$ \mathbf{w} := \nabla u; $$ then $\mathbf{w} \in \mathbf{L}^2(\Omega) $ since $u \in H^1(\Omega)$, $\operatorname{curl} \mathbf{w} = \mathbf{0}$ since $\mathbf{w}$ is a gradient and $\operatorname{div} \mathbf{w} = \operatorname{div}(\nabla u)= \Delta u = 0$ by construction.

On the other hand, it can happen that $\mathbf{w} \notin (H^1(\Omega))^3$, because $u$ is not necessarily in $H^2(\Omega)$: see remark above.

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  • $\begingroup$ Thank you very much. This was very helpful! $\endgroup$ – taylor123123 Apr 14 at 0:04
  • $\begingroup$ maybe you have also an idea to my very similar question math.stackexchange.com/questions/3186845/… $\endgroup$ – taylor123123 Apr 14 at 0:09
  • $\begingroup$ I will have a look at the other question. If my aswer was actually helpful, could you be so kind to accept it? $\endgroup$ – GaC Apr 15 at 12:06
  • $\begingroup$ I have accepted it! I have reformulated my other question here:math.stackexchange.com/questions/3188098/…. This gets better to the heart of the issue. $\endgroup$ – taylor123123 Apr 15 at 14:43
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The equation (5.2) from above is indeed the example, I was looking for. See for example the book of Peter Monk on Finite Element Methods for Maxwell's equation on page 76 for the explanation.

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