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Mar 25 at 7:32 comment added Zhang Yuhan It seems the boundedness of the domain important for the equivalence. For if $\Omega = \mathbb R^n$ the space $L^2_0(\Omega)$ would be undefined and the range of $\operatorname{div}: H_0^1(\Omega)\to L^2(\Omega)$ would not be closed. I wonder are there proofs of J.C. Lions lemma that makes no use of Fourier analysis when $\Omega$ unbounded.
Jan 13 at 14:08 vote accept Zhang Yuhan
Jan 13 at 14:00 history edited Carlo Beenakker CC BY-SA 4.0
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Jan 13 at 13:35 history answered Carlo Beenakker CC BY-SA 4.0