Poincaré inequality

Remarks

For part 1 one can take $U$ inside a slab $\set{0 < x_1 < d}$. For smooth $u$ with compact support, $u(x) = \int_0^{x_1} \partial_1 u \, dt$, and Hölder's inequality gives $\norm{u}_{L^p} \le d \norm{\partial_1 u}_{L^p}$; density extends this to $W^{1,p}_0(U)$. Part 2 is proved by contradiction. If it failed, there would be functions with mean $0$, $L^p$ norm $1$ and gradients tending to $0$. The Rellich–Kondrachov theorem would give a limit with zero gradient, hence constant on the connected set $U$. This limit has mean $0$, so it is $0$, which contradicts the norm $1$.

The hypotheses are needed. In part 2 one must subtract the mean, since constant functions have zero gradient. Connectedness is needed: on two disjoint intervals, a function equal to $1$ on one and $-1$ on the other has mean $0$, zero gradient and nonzero norm.

Going deeper

For a student

The inequality shows that $\norm{\nabla u}_{L^2}$ is a norm on $H^1_0(U)$ equivalent to the full $H^1$ norm. This is exactly the coercivity needed to apply the Lax–Milgram theorem to the Dirichlet problem for $-\Delta u = f$. The best constant for $p = 2$ in part 1 is $1 / \sqrt{\lambda_1}$, where $\lambda_1$ is the first Dirichlet eigenvalue of $-\Delta$ on $U$.