 Solution

The appropriate Poincare-Wirtinger inequality, with no zero-boundary assumption, isAssume $\Omega$ is nonempty, bounded, connected and has the stated cone property. If this failed, after subtracting the mean and normalizing there would be $u_j\in W^{1,p}(\Omega )$ withThe sequence is bounded in $W^{1,p}$. By the permitted Rellich-Kondrachov compactness theorem, a subsequence converges strongly in $L^p$ to $u$. For every compactly supported test function, integration by parts and these two convergences show that each distributional derivative of $u$ is zero. A function with zero weak gradient is constant on a connected open set: mollification makes it locally constant, and connectedness identifies the local constants.

Convergence in $L^p$ on a finite-measure set also preserves the integral, so $u_{\Omega }=0$ and the constant is zero. But strong convergence preserves $\Vert u{\Vert }_p=1$, a contradiction. This proves the inequality; connectedness prevents different constants on separate components.