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)?
2026-05-15 16:00:36.1778860836
Sobolev trace condition
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