0
$\begingroup$

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!

Improve this question
$\endgroup$
2
  • $\begingroup$ Is there something missing from the problem? First of all, there is no coupling between the two integrals, so you could equally first take the supremum of one and then add it to the other. Secondly there are no additional bounds and the problem is linear, so the solution will always be either 0 or $\infty$. $\endgroup$
    – mlk
    Commented Feb 18, 2022 at 14:56
  • $\begingroup$ @mlk my bad, there was some information missing indeed. I have modified the question, assuming there is just $E$ and adding the constraint. $\endgroup$
    – JaberEdgar
    Commented Feb 21, 2022 at 17:46

0

Reset to default

You must log in to answer this question.