Timeline for On Talenti's proof of optimal constant in Sobolev inequality
Current License: CC BY-SA 4.0
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| Dec 20, 2023 at 18:57 | comment | added | Deane Yang | Finally, there is a hybrid proof, where you first use symmetrization to reduce it to a $1$-dimensional variational problem. That problem can be solved using a $1$-dimensional version of the CNV proof. In particular, you can use the appropriate moment-entropy and Fisher information-entropy inequalities proved here: math.nyu.edu/~yangd/papers/renyi1d.pdf | |
| Dec 20, 2023 at 18:54 | comment | added | Deane Yang | A beautiful proof that I find easier to understand is by Cordero, Nazaret, Villani. webusers.imj-prg.fr/~dario.cordero/Docs/articles/CNV.pdf. It is based on optimal mass transport, but you don't need to know anything except for the existence of what's known as the Brenier map. | |
| Dec 20, 2023 at 18:52 | comment | added | Deane Yang | Aubin's proof was originally published here: Aubin T Problemes isoperimetriques et espaces de Sobolev. J. Diff. Geo. 11. (1976) 573. It also appears in his book Nonlinear Analysis on Manifolds. Monge-Ampère Equations. | |
| Dec 20, 2023 at 17:44 | comment | added | user519428 | If you could give me some references, that would be very great, thank you! | |
| Dec 20, 2023 at 15:41 | comment | added | Deane Yang | This doesn't really answer your question but there are simpler proofs of this using mass transport. I was never able to work through Talenti's proof. Aubin also has a similar proof, which you could also look at. I can provide references if you're interested. | |
| Dec 20, 2023 at 14:22 | history | edited | Daniele Tampieri | CC BY-SA 4.0 |
Formatting and minor Math Jaxing
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| S Dec 20, 2023 at 13:37 | review | First questions | |||
| Dec 20, 2023 at 14:22 | |||||
| S Dec 20, 2023 at 13:37 | history | asked | user519428 | CC BY-SA 4.0 |