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)



A Poincare inequality for the Laplace-Beltrami operator
Here is an elementary paper of Seeley that might help


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


