Poincare-Wirtinger inequality

On a bounded connected Lipschitz domain $U$, every $u\in H^1(U)$ satisfiesEquivalently, the gradient norm is equivalent to the $H^1$ norm on the mean-zero Sobolev space.

For the cube $Q=(0,L{)}^n$ and $u\in H^1(Q)$,This follows from the pairwise-difference identityand a coordinate-by-coordinate path from $x$ to $y$. The Cauchy-Schwarz inequality bounds the squared sum of its $n$ increments by $n$ times the sum of their squares, while the one-dimensional fundamental theorem of calculus bounds each integrated increment by $L^{n+2}\Vert D_{i}u{\Vert }_2^2$. Division by $2|Q|=2L^n$ gives the stated constant.

The mean-zero Sobolev space is the closed subspaceOn a bounded connected Lipschitz domain, the Poincare-Wirtinger inequality makes $\Vert \nabla u{\Vert }_2$ an equivalent Hilbert norm on this space.