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Suppose $w \in C^2 (S^{n-1}), \Lambda$ is Laplace-Beltrami operator on the sphere $S^{n-1}$, How can I prove follow Poincare inequality : $\int_{S^{n-1}} w\Lambda w d\sigma \leq (1-n) \int_{S^{n-1}} |w|^2 d\sigma$

Remark: Set $w(x)=w(r,\theta), (r, \theta)$ are the polar coordinates in $R^n$, we get the following formula $\Delta_x w(x)=\frac{\partial^2u}{\partial r^2}+\frac{n-1}{r}+\frac{1}{r^2}\frac{\partial u}{\partial r}+\frac{1}{r^2}\Lambda w(r,\theta)$, See Page 181 of the Aviles' article: Local Behavior of Solutions of Some Elliptic Equations, Communications Mathematical Physics, 108, 177-192(1987) . also see page 34 of Veron's article : Comportement asymptotique des solutions d'equations elliptiques semi-lineaires dans $R^N$ , Ann.Mat.Pura Appl CXXVII,25-50(1981)

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    $\begingroup$ You need to define $\Lambda$ precisely because tehre are two different sign conventions, depending on whether you are a geometer or not. I assume you are not, so your Laplacian is negative semidefinite. As is, the inequality is not true. Suppose that $w=1$. The left hand side is $0$ while the right-hand side is negative for $n>1$. The inequality has to do with the first nonzero eigenvalue of the Laplacian on the round $(n-1)$-sphere which is $\pm(n-1$ depending on your conventions for $\Lambda$. Try wikipedia en.wikipedia.org/wiki/Spherical_harmonics $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 6, 2013 at 13:23
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    $\begingroup$ The question you asked is a simple exercise involving Ritz-Raleigh quotients, taking into accound that the first nozero e-value of $\Lambda$ is $(1-n)$. The inequality you're looking for states that $\int_{S^{n-1}} w \Lambda w dS \leq (1-n)\int_{S^{n-1}} w^2 dS$ provided that $\int_{S^{n-1}} w=0$. As explained in my previous comment, the inequality is not true if the mean of $w$ is not zero. $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 6, 2013 at 14:07
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    $\begingroup$ I gave you a wikipedia link. Here it is again en.wikipedia.org/wiki/Spherical_harmonics $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 6, 2013 at 14:42
  • $\begingroup$ Thanks, I don't know the first nonzero e-value of $\Lambda$ is $(1-n)$. In fact Veron's paper has said that" Sachant que N-1 est la seconde valeur propre de -$\Lambda$", But I can't understand French.At last I translate the sentence to English by Google.But Why the first nonzero e-value of $\Lambda$ is (1-n), Can you point out some references for me? $\endgroup$
    – bigheadliao
    Commented Nov 6, 2013 at 14:45
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    $\begingroup$ This is the third tiome I am giving you the Wikipedia link where you can find the result about the eigenvalues of $\Lambda$. All you have to do is click on it. en.wikipedia.org/wiki/Spherical_harmonics $\endgroup$
    – Liviu Nicolaescu
    Commented Nov 6, 2013 at 15:50

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Here is an elementary paper of Seeley that might help

http://www.jstor.org/stable/2313760

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