# Periodic Poincaré Inequality?

The classical periodic Poincaré inequality states that if $u\in H^1(\mathbb T^n)$ is such that $\displaystyle\int_{\mathbb T^n} u(x)\ dx=0$ then $$\|u\|_{L^2(\mathbb T^n)}^2\leq C_d \|\nabla u\|_{L^2(\mathbb T^n)}^2,$$ for some constant $C$.

Can one drop the hypothesis $\displaystyle \int_{\mathbb T^n} u(x)\ dx=0$?

As literally stated? No, we can't. (I'm taking the volume of the torus to be 1 for convenience.) We can write any function as $u+c$ where $\int u = 0$; and replacing $u$ in your formula with $u+c$ the left hand side becomes $\|u\|^2_{L^2} + c\int u(x)dx + c^2 = \|u\|^2_{L^2} + c^2$. Then we obviously cannot bound this by $C_d\|\nabla (u+c)\|^2_{L^2}=C_d\|\nabla u\|^2_{L^2}$ because I can make $c$ arbitrarily large. But as long as you add a factor to counteract this on the RHS like, say, $$\|u\|^2_{L^2} \leq C_d \|\nabla u\|^2_{L^2} + \left(\int u(x)dx\right)^2,$$ then it's true.