# Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Hello, I asked this question before, but didn't get any response, so I took the liberty of asking once again , with slightly modified version of the question:

Consider an orientation-preserving quasiconformal homeomorphism $f$ of the open unit disk $\mathbb{D}\subset \mathbb{C}$ with the complex dilatation/Beltrami cofficient $\mu, ||\mu||_{L^\infty(\mathbb{D})}<1, \mu \in C^0(\mathbb{\bar{D}})$ , i.e. $\mu$ is continuous on the closed unit disk $\mathbb{\bar{D}}$. Is it true that the restriction of $f$ on the boundary $S^1$ has continuous (ordinary) derivative on $S^1$, i.e., is $f\in C^1(S^1)$ ?

I guess one might start with continuously extending $\mu$ to all of $\mathbb{C}$, then solve the Beltrami equation on $\mathbb{C}$ with that extended $\mu$, but then I am getting stuck:because I guess the solution to this new equation might not be $C^1(\mathbb{C})$ ?? ( Look at Examples 15.1 in the book "Elliptic PDE and Quasiconformal Mappings" by K. Asltala, T. Iwaniec and G. Martin.

May be, to start with, we can ALSO assume that $\mu=0$on $S^1$. Then is the answer to my question yes ?

Could you please cite a reference of your proof to my question or give a counterexample ? Thanks a lot !!

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IMHO, either this or mathoverflow.net/questions/79337/… should be closed, and answers/comments left on the remaining one –  Yemon Choi Jun 21 '12 at 2:21
@ Yemon Choi : these two questions are different, in the other question which I asked a while back, I was looking for $C^1(\bar{D})$ regularity, which I already got the counter examples of. Here I am asking a weaker question, namely, whether the restriction of $f$ on $S^1$ is $C^1$, so I am not looking at the $C^1$ regularity upto the boundary, I am only looking whether the restriction of the solution to the boundary is $C^1.$ P.S. I already deleted the question ( with the same content ) which I asked two days back. –  Analysis Now Jun 21 '12 at 2:29
Sorry; my mistake. –  Yemon Choi Jun 21 '12 at 4:35
I have two suggestions: 1. Take a look at the linearization of the Beltrami equation, $v_{\bar{z}}=\nu$, where $v$ is a vector field. The point is that linearly interpolating from $\mu$ to $0$ you get a 1-parameter family of qc maps, differentiating this family you get time-dependent vector fields. Thus, positive answer in linear case could lead to positive answer to your question. 2. MO is clearly a wrong place for this question since nobody here is doing hard-core qc analysis. Write email to few experts, I think, you know who they are, and see what they say. –  Misha Jun 22 '12 at 0:43
Dear Prof. Kapovich, thanks for your comment. 1) What did you mean by linearization of the Beltrami equation, because Beltrami equation $f_{\bar{z}}= \mu. f_z$ is a linear equation in $f$, if $\mu$ is taken to be the same in each case. 2) I has e-mailed to people whom I knew are pioneers in this area, but unfortunately that was not of much help to me. –  Analysis Now Jul 10 '12 at 4:04

They also mention the example $$f(z) = z\cdot (1-\log|z|),$$ which defines a quasiconformal function near zero whose complex dilatation is continuous near zero, but which is not differentiable at zero.