0
$\begingroup$

Let $f:\mathbb{S}^1 \to \mathbb{S}^1$ be an orientation-preserving homeomorphism. Denote by $H(f)$ the complex harmonic extension/solution in $\mathbb{D}$ to the Dirichlet problem with boundary data $f$, which can also be shown to be a homeomorphism of $\bar{\mathbb{D}}$ [Choquet's theorem].

Now,it follows from a well-known theorem of Kellog-Warschowski that if $f\in C^{k,\alpha}(\mathbb{S^1})$, then $H(f) \in C^{k,\alpha}(\mathbb{D})$.

My question is : assuming $k=1$, what is the limiting matrix of the (total) derivative matrix $DH(f)_p$ as $p\to \zeta \in \mathbb{S^1},p \in \mathbb{D}$ ? From some other kind of extensions I have seen before, I would take a guess that $DH(f)_p \to f'(\zeta).Id$ as $p\to \zeta$, where $Id$ denotes the $2$ by $2$ identity matrix.

A reference to your answer will be highly appreciated!

Improve this question
$\endgroup$

1 Answer 1

Reset to default
1
$\begingroup$

Your guess is wrong. Take real $f$, for example, then harmonic extension is also real, and its derivative is singular. In general, I do not expect a simple answer. And certainly the answer cannot depend on the local properties of $f$ near $\zeta$ only. Indeed, leave $f$ unchanged on one half of the circle, but change it on another half. It is clear that the derivative at the points of the first half will change. So the limit of the derivative at a point cannot be computed from the knowledge of f only near this point.

Improve this answer
$\endgroup$
7
  • $\begingroup$ 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 ? $\endgroup$
    – Analysis Now
    Commented Sep 24, 2012 at 4:55
  • $\begingroup$ Where is your modified question? I don't see it. What kind of extension do you want to consider now? $\endgroup$
    – Alexandre Eremenko
    Commented Sep 24, 2012 at 13:11
  • $\begingroup$ 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 $p \to \zeta, p \in \mathbb{D}, \zeta \in S^1$ ? $\endgroup$
    – Analysis Now
    Commented Sep 24, 2012 at 13:54
  • $\begingroup$ 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. $\endgroup$
    – Analysis Now
    Commented Sep 24, 2012 at 13:57
  • $\begingroup$ What is the difference between the "modified question" and the original question? I don't see any. $\endgroup$
    – Alexandre Eremenko
    Commented Sep 25, 2012 at 11:50

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .