= Poincare inequality with a partial Dirichlet boundary
{c}
If $U$ is a bounded connected <Lipschitz domain> and a measurable boundary portion $\Gamma_D$ has positive surface measure, then
$$
\|v\|_{L^2(U)}\leq C\|\nabla v\|_{L^2(U)}
$$
for every $v\in H^1(U)$ whose <Sobolev trace operator>[trace] vanishes on $\Gamma_D$.
Otherwise choose $\|v_j\|_2=1$ with $\|\nabla v_j\|_2\to0$. The <Rellich-Kondrachov compactness theorem> for $H^1(U)$ gives a strongly $L^2$ convergent subsequence and a weakly $H^1$ convergent subsequence with the same limit. Its <weak gradient> is zero, so connectedness makes it constant. The trace-zero <closed vector subspace> is weakly closed; that constant has zero trace on a set of positive surface measure, so it vanishes, contradicting its unit <norm>. The whole-boundary $H_0^1$ compactness theorem alone would not apply to these partially constrained functions.
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