The $L^p$ Teichmüller theory: Existence and regularity of critical points

Abstract

We study minimisers of the $p$-conformal energy functionals, \[ \mathsf{E}_p(f):=\int_\ID \IK^p(z,f)\,dz,\quad f|_\IS=f_0|_\IS, \] defined for self mappings $f:\ID\to\ID$ with finite distortion and prescribed boundary values $f_0$. Here \[ \IK(z,f) = \frac{\|Df(z)\|^2}{J(z,f)} = \frac{1+|μ_f(z)|^2}{1-|μ_f(z)|^2}\] is the pointwise distortion functional and ${\mu }_f(z)$ is the Beltrami coefficient of $f$. We show that for quasisymmetric boundary data the limiting regimes $p\to \infty$ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for $p\to 1$ recovers the harmonic mapping theory. Critical points of ${\mathsf{E}}_p$ always satisfy the inner-variational distributional equation \[ 2p\int_\ID \IK^p\;\frac{\overline{μ_f}}{1+|μ_f|^2}φ_\zbar \; dz=\int_\ID \IK^p \; φ_z\; dz,\quad\forallφ\in C_0^\infty(\ID ). \] We establish the existence of minimisers in the {\em a priori} regularity class $W^{1,\frac{2p}{p+1}}(\ID)$ and show these minimisers have a pseudo-inverse - a continuous $W^{1,2}(\ID)$ surjection of $\ID$ with $(h\circ f)(z)=z$ almost everywhere. We then give a sufficient condition to ensure $C^{\infty}(\ID)$ smoothness of solutions to the distributional equation. For instance $\IK(z,f)\in L^r_{loc}(\ID)$ for any $r>p+1$ is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further $\IK(w,h)\in L^1(\ID)$ will imply $h$ is a homeomorphism, and together these results yield a diffeomorphic minimiser. We show such higher regularity assumptions to be necessary for critical points of the inner variational equation.