Timeline for A Poincare inequality for the Laplace-Beltrami operator [closed]
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| S Nov 7, 2013 at 8:45 | history | unlocked | CommunityBot | ||
| S Nov 7, 2013 at 8:45 | history | locked | CommunityBot | ||
| S Nov 7, 2013 at 8:45 | history | closed |
Benoît Kloeckner Stefan Kohl♦ Chris Godsil David White Willie Wong |
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| Nov 6, 2013 at 19:04 | review | Close votes | |||
| Nov 7, 2013 at 8:45 | |||||
| Nov 6, 2013 at 18:42 | history | edited | Stefan Kohl♦ | CC BY-SA 3.0 |
Corrected spelling of the title.
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| Nov 6, 2013 at 16:17 | vote | accept | bigheadliao | ||
| Nov 6, 2013 at 16:05 | answer | added | Liviu Nicolaescu | timeline score: 0 | |
| Nov 6, 2013 at 15:50 | comment | added | Liviu Nicolaescu | 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 | |
| Nov 6, 2013 at 14:45 | comment | added | bigheadliao | 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? | |
| Nov 6, 2013 at 14:42 | comment | added | Liviu Nicolaescu | I gave you a wikipedia link. Here it is again en.wikipedia.org/wiki/Spherical_harmonics | |
| Nov 6, 2013 at 14:07 | history | edited | bigheadliao | CC BY-SA 3.0 |
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| Nov 6, 2013 at 14:07 | comment | added | Liviu Nicolaescu | 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. | |
| Nov 6, 2013 at 13:47 | history | edited | bigheadliao | CC BY-SA 3.0 |
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| Nov 6, 2013 at 13:23 | comment | added | Liviu Nicolaescu | 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 | |
| Nov 6, 2013 at 13:01 | history | asked | bigheadliao | CC BY-SA 3.0 |