The Teichmüller problem for $L^p$means of distortion
Abstract
Teichmüller's problem from 1944 is this: Given $x\in [0,1)$ find and describe the extremal quasiconformal map $f:\ID\to\ID$, $f\partial \ID=identity$ and $f(0)=x\leq 0$. We consider this problem in the setting of minimisers of $L^p$mean distortion. The classical result is that there is an extremal map of Teichmüller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x\neq 0$. For the $L^p$norm, when $p=1$ it is known that there can be no locally quasiconformal minimiser unless $x=0$. Here we show that for $1\leq p<\infty$ there is a minimiser in a weak class and an associated AhlforsHopf holomorphic quadratic differential with a pole of order $1$ at $f(0)=r$. However, this minimiser cannot be in $W^{1,2}_{loc}(\ID)$ unless $r=0$ and $f=identity$. Hence there is no locally quasiconformal minimiser. A similar statement holds for minimsers of the exponential norm of distortion. We also use our earlier work to show that as $p\to\infty$, the weak $L^p$minimisers converge locally uniformly in $\ID$ to the extremal quasiconformal mapping, and that as $p\to 1$ the weak $L^p$minimisers converge locally uniformly in $\ID$ to the identity.
 Publication:

arXiv eprints
 Pub Date:
 July 2021
 arXiv:
 arXiv:2107.07660
 Bibcode:
 2021arXiv210707660M
 Keywords:

 Mathematics  Complex Variables