If we have two functions $f,g \in W^{1,m}(E)$ in some domain $E\subset\mathbb{R}^n$ who have a trace on $\partial E$ with $f\leq g$ on $\partial E$. Is it then always possible to still have $f>g$ everywhere in $E$ ($f$, $g$ not continuous)?
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2Not if $m$ is so large that $W^{1, m}\subset C^0$, obviously. – Giuseppe Negro Dec 28 '22 at 11:31
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What do you mean by is it always possible? For example, if it happens that $f \equiv g$, then no, it's not possible to have $f > g$. – Michał Miśkiewicz Dec 31 '22 at 06:13