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May 12, 2019 at 15:48 comment added fedja The task is, apparently, to minimize $\int_\Omega|\nabla u|^2$ and to show that the minimizer exists, which more or less amounts to showing that $K$ is closed in $H^1$. (BTW, the condition $u\ge\varphi$ on $E$ should be understood with certain care because point evaluation is not continuous in $H^1$ when $n\ge 2$).
May 12, 2019 at 14:47 history asked yoshi CC BY-SA 4.0