= Poincare inequality with a boundary trace
{c}
On a bounded connected Lipschitz domain $U$,
$$
\|u\|_{L^2(U)}\leq C\left(\|Du\|_{L^2(U)}+\|\operatorname{Tr}u\|_{L^2(\partial U)}\right)
$$
for every $u\in H^1(U)$. If this failed, the <Rellich-Kondrachov compactness theorem> would produce a normalized strong $L^2$ limit with zero <gradient>, hence a <constant function>; continuity of the <Sobolev trace theorem> would force that constant to vanish, contradicting its unit <norm>.
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