= Poincare inequality with a partial Dirichlet boundary
{c}
If $U$ is bounded and connected and a boundary portion $\Gamma_D$ has positive surface measure, then
$$
\lVert v\rVert_{L^2(U)}\leq C\lVert\nabla v\rVert_{L^2(U)}
$$
for every $v\in H^1(U)$ whose <Sobolev trace theorem>[trace] vanishes on $\Gamma_D$. Otherwise a normalized counterexample sequence, the <Rellich-Kondrashov compactness theorem for H01>[Rellich-Kondrashov compactness theorem], and continuity of the trace would produce a nonzero constant with zero trace on $\Gamma_D$.
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