The Teichmuller problem for $L^p$-means of distortion

Teichmullers problem from 1944 is this: Given $xin [0,1)$ find and describe the extremal quasiconformal map $f:IDtoID$, $f|partial ID=identity$ and $f(0)=-xleq 0$. 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 Teichmuller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x eq 0$. 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 $1leq p<infty$ there is a minimiser in a weak class and an associated Ahlfors-Hopf 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 $ptoinfty$, the weak $L^p$-minimisers converge locally uniformly in $ID$ to the extremal quasiconformal mapping, and that as $pto 1$ the weak $L^p$-minimisers converge locally uniformly in $ID$ to the identity.