Timeline for What is the limit of the derivative of the harmonic extension/Dirichlet solution in $C^{1,\alpha}$ cases?
Current License: CC BY-SA 3.0
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| Sep 26, 2012 at 2:23 | comment | added | Alexandre Eremenko | But this is the same thing: changing your homeomorphism near the point -1, for example will change the derivative of the harmonic extension near 1. The derivative of the harmonic extension near a point does not only depend on your boundary function near this point, but depends on the whole boundary function. So you cannot possibly hope for anything like $DH(f)\to f'.Id$. | |
| Sep 25, 2012 at 13:47 | comment | added | Analysis Now | The previous question was $f:S^1\to C$,and this function was any $C^{1,\alpha}$ function on $S^1$, and in answering, you chose the particular case where $f:S^1\to R$. I changed the question to where $f:S^1\to S^1$ is any orientation-preserving $C^{1,\alpha}$ diffeomorphism of $S^1$, so we cannot consider $f:S^1\to R$ anymore. Does that seem clear ? Thanks. | |
| Sep 25, 2012 at 11:50 | comment | added | Alexandre Eremenko | What is the difference between the "modified question" and the original question? I don't see any. | |
| Sep 24, 2012 at 13:57 | comment | added | Analysis Now | The other extension I am talking about comes from theory of Riemann surfaces, where there is a particular kind of extension called Douady-Earle extension $F(f)$ where $DF(f)_p \to f'(\zeta)$ as $p \to \zeta$, where $f$ is a $C^{1,\alpha}$ circle diffeomorphims (actually $C^1$ suffices. | |
| Sep 24, 2012 at 13:54 | comment | added | Analysis Now | The modified question is : $f:S^1\to S^1$ is now a $C^{1,\alpha}$ diffemorphism. I am still considering its complex harmonic extension $H(f)$ and am asking whether $DH(f)_p \to f'(\zeta).Id$ as $p \to \zeta, p \in \mathbb{D}, \zeta \in S^1$ ? | |
| Sep 24, 2012 at 13:11 | comment | added | Alexandre Eremenko | Where is your modified question? I don't see it. What kind of extension do you want to consider now? | |
| Sep 24, 2012 at 4:55 | comment | added | Analysis Now | Thanks for your answer, but I needed to change the question to suit the particular cases that is required for me. I know other type of extensions (not harmonic extension) where the limit depends on $\zeta$ only.Do you think in my modified question, there is a chance that the limit should be as I mentioned ? | |
| Sep 24, 2012 at 3:55 | history | answered | Alexandre Eremenko | CC BY-SA 3.0 |