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!

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

If we take $E$ with regular boundary then, the problem consists in maximizing the integral on $E$ of the curl operator. I think here that with the help of the Stokes formula the problem is well defined.

Is there anything we can say if $E$ is of finite perimeter for instance?

Thanks!