Timeline for Is the universal constant in Caccioppoli's inequality one?
Current License: CC BY-SA 3.0
12 events
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| Jul 8, 2014 at 18:31 | comment | added | username | @NateEldredge indeed. | |
| Jul 8, 2014 at 16:26 | comment | added | Nate Eldredge | The last inequality in your displayed proof is actually an equality, right? | |
| Jul 8, 2014 at 5:47 | comment | added | username | @AnthonyQuas I have added it in. | |
| Jul 8, 2014 at 5:46 | history | edited | username | CC BY-SA 3.0 |
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| Jul 8, 2014 at 2:43 | comment | added | Christian Remling | @AnthonyQuas: This doesn't hold pointwise; we need to integrate by parts and use the assumption that $\Delta u\ge 0$. | |
| Jul 8, 2014 at 1:32 | comment | added | Anthony Quas | Why is $|\nabla(u\chi)|\le |u||\nabla\chi|$? | |
| Jul 7, 2014 at 21:33 | comment | added | Deane Yang | I'm not so familiar with Caccioppoli inequalities, but for an example where getting a constant less than 1 for a similar estimate really matters, see Schoen, Simon, and Yau's paper, "Curvature estimates for minimal hyper surfaces". | |
| Jul 7, 2014 at 21:17 | comment | added | username | @DeaneYang Thank you. Caccioppoli is often used iteratively, so any number larger than one blows up. Is the optimal constant possibly smaller than one? | |
| Jul 7, 2014 at 21:08 | comment | added | Deane Yang | This practice is rather common. Usually such an inequality is but one in a long chain of inequalities, where the exactly formula for the constant becomes more complicated but its exact formula is of no consequence to the proof. So a constant $C$ is used, whether it is really $1$ or something more elaborate. Also, the use of the constant $C$ signals more clearly that its exact value doesn't matter. That said, sometimes using the exact value of a constant does lead to useful stronger results that are sometimes missed by people who aren't keeping careful track of the constants. | |
| Jul 7, 2014 at 20:59 | history | edited | username | CC BY-SA 3.0 |
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| Jul 7, 2014 at 20:33 | history | edited | username | CC BY-SA 3.0 |
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| Jul 7, 2014 at 20:27 | history | asked | username | CC BY-SA 3.0 |