The Teichm\"uller problem for $L^p$-means of distortion

Teichm"uller’s problem from 1944 is this: Given $x\in [0,1)$ find and describe the extremal quasiconformal map $f:\text{ID}\to \text{ID}$, $f|\partial \ID=identity$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 Teichm"uller type with associated holomorphic quadratic differential having a pole of order one at $x$, if $x\mathrm{\ne }0$. 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 $1\le p<\mathrm{\infty }$ 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 $W_{loc}^{1,2}(\text{ID})$ 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 $p\to \mathrm{\infty }$, the weak $L^p$-minimisers converge locally uniformly in $\text{ID}$ to the extremal quasiconformal mapping, and that as $p\to 1$ the weak $L^p$-minimisers converge locally uniformly in $\text{ID}$ to the identity.