Function in H(curl) $\cap$ H(div), but not in H1

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.

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.

• Thank you very much. This was very helpful! – taylor123123 Apr 14 at 0:04
• maybe you have also an idea to my very similar question math.stackexchange.com/questions/3186845/… – taylor123123 Apr 14 at 0:09
• I will have a look at the other question. If my aswer was actually helpful, could you be so kind to accept it? – GaC Apr 15 at 12:06
• 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. – taylor123123 Apr 15 at 14:43

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.