Boundary regularity of quasiconformal homeomorphisms of the unit disk ? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:39:49Z http://mathoverflow.net/feeds/question/100198 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/100198/boundary-regularity-of-quasiconformal-homeomorphisms-of-the-unit-disk Boundary regularity of quasiconformal homeomorphisms of the unit disk ? Analysis Now 2012-06-21T01:40:35Z 2012-11-03T11:09:59Z <p>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:</p> <p>Consider an orientation-preserving quasiconformal homeomorphism $f$ of the open unit disk $\mathbb{D}\subset \mathbb{C}$ with the complex dilatation/Beltrami cofficient $\mu, ||\mu||_{L^\infty(\mathbb{D})}&lt;1, \mu \in C^0(\mathbb{\bar{D}})$ , 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 $f\in C^1(S^1)$ ?</p> <p>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.</p> <p><a href="http://books.google.com/books?id=5aOgM9XRiXUC&amp;printsec=frontcover&amp;hl=sl#v=onepage&amp;q&amp;f=false" rel="nofollow">http://books.google.com/books?id=5aOgM9XRiXUC&amp;printsec=frontcover&amp;hl=sl#v=onepage&amp;q&amp;f=false</a></p> <p>May be, to start with, we can ALSO assume that $\mu=0$on $S^1$. Then is the answer to my question yes ?</p> <p>Could you please cite a reference of your proof to my question or give a counterexample ? Thanks a lot !!</p> http://mathoverflow.net/questions/100198/boundary-regularity-of-quasiconformal-homeomorphisms-of-the-unit-disk/111362#111362 Answer by Lasse Rempe-Gillen for Boundary regularity of quasiconformal homeomorphisms of the unit disk ? Lasse Rempe-Gillen 2012-11-03T11:09:59Z 2012-11-03T11:09:59Z <p>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.</p> <p>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. </p> <p>Note that this example is real on the real axis, so it easily yields a counterexample to your question (unless I have misunderstood something!).</p>