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Mar 11, 2014 at 10:01 comment added username you just need to show that the minimiser you obtain is in $W^{1,4}$. Note that $\mathcal{E}$ is not closed, so it is an infimum and not a minimum in general, cf. detailed answer above.
Mar 11, 2014 at 9:32 history answered MSSHD CC BY-SA 3.0