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Let $f \in H^{-1}(S)$, where $S \subset \mathbb{R}^n$ is some nice set in space. Can I exchange supremums and integrals over time here: $$\sup_{g \in H^1(S)}\int_0^T \langle f, g \rangle_{H^{-1}(S), H^1(S)}\;dt =\int_0^T \sup_{g \in H^1(S)} \langle f, g \rangle_{H^{-1}(S), H^1(S)}\;dt$$

I think so. But what if $g$ has some dependence on $t$? Or is that not allowed?

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    What is the relation between $f$ and $y$?2012-12-12
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    @HansEngler Sorry it should all be $f$. I edited.2012-12-12
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    ... and $S = \Omega$?2012-12-12
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    @HansEngler $S = \Omega$, some nice set. No time dependence.2012-12-12

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