Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 105 / 2 / b

If the asserted Poincaré inequality failed, after rescaling there would be $u_j\in H$ withThe sequence is bounded in $H^1(U)$. By the Rellich-Kondrachov compactness theorem, a subsequence converges strongly in $L^2(U)$ to some $u$ and weakly in $H^1(U)$. Its weak gradient is zero, so connectedness of $U$ makes $u$ almost everywhere constant. Continuity of the integral under $L^2$ convergence gives ${\int }_{U}u=0$, hence $u=0$. Strong convergence would then imply $\Vert u_j{\Vert }_2\to 0$, contradicting the normalization. Therefore the required constant exists.