I am interested in the following Poincaré-type inequality,
$$ \int\_{S(r)} \lvert u-\bar{u}\rvert^2 d\sigma \leq C(N) \int\_{S(r)} |u\_{\theta}|^2 d\sigma$$
where $\bar{u} = \frac{1}{\lvert S(r)\rvert}\int\_{S(r)} u d\sigma$ and $u\_{\theta}$ denotes the tangential derivative of $u$. The domain $S(r)$ is just the $N$-dimensional sphere with radius $r$. The function $u$ can be assumed to be smooth for the purpose of this question.


In the case $S(r)$ is $1$ dimensional (i.e. a circle), then the constant $C(1)=1$ and this is the Wirtinger's inequality. Are there any references where I can find the best constant in higher dimensions?



Best constant for Poincaré inequality on spheres
The best constant is just the multiplicative inverse of the smallest positive eigenvalue of the Laplacian on the sphere. On $\mathbb{S}^N$ this the smallest eigenvalue is $N$, so $C(N)$ in that case equals $1/N$.


You can figure out the appropriate $r$ scaling yourself.


