Timeline for Best constant for Poincaré inequality on spheres
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 11, 2022 at 21:17 | comment | added | Adi | Thanks for the information, i was searching on Google, but couldn't find any relevant estimates except in 1D. | |
| Dec 11, 2022 at 15:24 | comment | added | Willie Wong | 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. | |
| Dec 10, 2022 at 18:36 | comment | added | Adi | Do you by any chance know of a result that proves the best consrant with 2 replaced by p? | |
| Mar 11, 2022 at 7:05 | comment | added | Adi | The book by Stein and Weiss is perfect, that's what I was looking for. | |
| Mar 11, 2022 at 7:04 | comment | added | Adi | Thanks very much for the clarification. I thought one could somehow use Rayleigh quotient, was not aware of this way of computing the explicit eigen functions. | |
| Mar 10, 2022 at 15:24 | comment | added | Willie Wong | 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. | |
| Mar 10, 2022 at 15:13 | comment | added | Willie Wong | @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. | |
| Mar 10, 2022 at 9:29 | comment | added | Adi | 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? | |
| Mar 10, 2022 at 9:27 | vote | accept | Adi | ||
| Mar 9, 2022 at 21:15 | comment | added | Willie Wong | 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. | |
| Mar 9, 2022 at 21:10 | history | answered | Willie Wong | CC BY-SA 4.0 |