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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?

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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.

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    $\begingroup$ Incidentally, this fact is generally true. If you have a closed connected Riemannian manifold, the global Poincare inequality like you stated has the best constant equal to the inverse of smallest positive eigenvalue of the Laplace-Beltrami operator (with sign condition so the spectrum is non-negative). If you have a compact connected manifold with boundary, the principal eigenvalue of the Dirichlet Laplacian provides the best constant for the Poincare inequality $\|u\|_{L^2}^2 \lesssim \|\nabla u\|_{H^1}^2$ for functions vanishing on the boundary. $\endgroup$
    – Willie Wong
    Commented Mar 9, 2022 at 21:15
  • $\begingroup$ Thanks for your reply, I know that the best Poincare inequality constant is the inverse of the first eigenvalue, which follows from using the Rayleigh quotient. I was looking for some reference where to find more about this and where this calculation is written down. Do you by any chance know of a good reference for the spectrum of the spherical laplacian? $\endgroup$
    – Adi
    Commented Mar 10, 2022 at 9:29
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    $\begingroup$ @Adi: Wikipedia? en.wikipedia.org/wiki/Spherical_harmonics#Higher_dimensions If you want detailed proofs and a citable source, try E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces (Princeton University Press). Chapter IV, Section 2. $\endgroup$
    – Willie Wong
    Commented Mar 10, 2022 at 15:13
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    $\begingroup$ The determination of the eigenvalue for the sphere usually doesn't do the Rayleigh quotient. One proves first that the spherical harmonics (as restrictions of harmonic polynomials on $\mathbb{R}^{N+1}$) are a complete system of orthogonal functions in $L^2$. That the polynomials are harmonic and homogeneous implies that the spherical harmonics are also eigenfunctions of the spherical Laplacian, with explicit eigenvalues. By the spectral theorem these spherical harmonics must be all the eigenvalues, and by inspection you see the smallest one. $\endgroup$
    – Willie Wong
    Commented Mar 10, 2022 at 15:24
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    $\begingroup$ Off the top of my head, I don't. But this would be equivalent to the eigenvalue problem for the p-Laplacian, which a brief google search shows there having a lot of research literature. $\endgroup$
    – Willie Wong
    Commented Dec 11, 2022 at 15:24

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