I am stuck at the following exercise: Let $\Omega \subset \mathbb{R}^2$. For $u \in H_{0}(\operatorname{curl},\Omega)$ there exists a decomposition $$u=\nabla \Phi +z$$ with $\Phi \in H_{0}^{1}$ and $z \in [H^{1}]^{2}$. Further $$\|\Phi\|_{H^{1}}+\|z\|_{H^{1}}\leq C\|u\|_{H(\operatorname{curl})}.$$
Up to this part I was successful, but I can't seem to be able to show the following second part: Now consider $\Omega=(0,1)\times (0,a)$ with $a\leq1$. Show that the constsnt $C$ must depend on $a$, i.e. $C\geq \frac{c}{a^2}$ with a new constant $c$ independent of $a$. The hint says to consider $$u(x,y):=(0,x(1-x))^{T}\in H_{0}(\operatorname{curl},\Omega).$$
My assumption is that the function in the hint should result in the inequality becoming an equality, but I wasn't able to work out the specific form of $\Phi$ in order to achieve the equality.
Edit:.$H_{0}(\operatorname{curl},\Omega):=\{ u\in [L^{2}]^2\: : \:\operatorname{curl}\in [L^{2}]^2, \operatorname{tr}(u)=u(x)\times n(x)=0\: \forall x\in\partial \Omega \}$, where $n$ denotes the outer normal and $\operatorname{tr}$ denotes the trace operator.