Poincare inequality with a partial Dirichlet boundary

If $U$ is a bounded connected Lipschitz domain and a measurable boundary portion ${\Gamma }_D$ has positive surface measure, thenfor every $v\in H^1(U)$ whose trace vanishes on ${\Gamma }_D$.

Otherwise choose $\Vert v_j{\Vert }_2=1$ with $\Vert \nabla v_j{\Vert }_2\to 0$. 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.