Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

2012-06-21T01:40:35Z

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.

http://books.google.com/books?id=5aOgM9XRiXUC&printsec=frontcover&hl=sl#v=onepage&q&f=false

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 !!

Answer by Lasse Rempe-Gillen for Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

2012-11-03T11:09:59Z

You may be interested in looking at the article "On boundary correspondence under quasiconformal mappings" by V.Ya. Gutlyanskii and V.I. Ryazanov. In particular, their Corollary 1 concerns what can be said when the Beltrami differential extends uniformly continuous to some part of the boundary.

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.

Note that this example is real on the real axis, so it easily yields a counterexample to your question (unless I have misunderstood something!).

Boundary regularity of quasiconformal homeomorphisms of the unit disk ? - MathOverflow

Boundary regularity of quasiconformal homeomorphisms of the unit disk ?

Answer by Lasse Rempe-Gillen for Boundary regularity of quasiconformal homeomorphisms of the unit disk ?