We fix $\Omega \subset \mathbb{R}^{2}$ an open set. My question is: what are the minimum conditions we need on $E,F\subset \Omega$ such that the following optimisation problem:
$$
\sup\{ \int_{F}\partial_{1}\psi_{2} - \int_{E}\partial_{2}\psi_{1}\; | \;(\psi_{1}, \psi_{2}) \in (\mathcal{C}^{1}_{0}(\Omega))^{2}\}
$$
admits a solution? 

If we take $E=F$ then, the problem consists in maximizing the integral on $E$ of the curl operator over $(\mathcal{C}^{1}_{0}(\Omega))^{2}$. I think here that with the help of the Stokes formula for sets with regular boundaries the problem is well defined. 

Is there anything we can say if $E\neq F$ are of finite perimeter for instance?

Thanks!