The Teichmüller problem for L^p-means of distortion

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