The \(L^p\) Teichmüller Theory: Existence and Regularity of Critical Points

Abstract

We study minimisers of the p-conformal energy functionals,

$$\begin{aligned} \textsf{E}_p(f):=\int _{\mathbb {D}}{\mathbb {K}}^p(z,f)\,\text {d}z,\quad f|_{\mathbb {S}}=f_0|_{\mathbb {S}}, \end{aligned}$$

defined for self mappings \(f:{\mathbb {D}}\rightarrow {\mathbb {D}}\) with finite distortion and prescribed boundary values \(f_0\). Here

$$\begin{aligned} {\mathbb {K}}(z,f) = \frac{\Vert Df(z)\Vert ^2}{J(z,f)} = \frac{1+|\mu _f(z)|^2}{1-|\mu _f(z)|^2} \end{aligned}$$

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 \(p\rightarrow \infty \) recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for \(p\rightarrow 1\) recovers the harmonic mapping theory. Critical points of \(\textsf{E}_p\) always satisfy the inner-variational distributional equation

$$\begin{aligned} 2p\int _{\mathbb {D}}{\mathbb {K}}^p\;\frac{\overline{\mu _f}}{1+|\mu _f|^2} \varphi _{\overline{z}}\; \text {d}z=\int _{\mathbb {D}}{\mathbb {K}}^p \; \varphi _z\; \text {d}z, \quad \forall \varphi \in C_0^\infty ({\mathbb {D}}). \end{aligned}$$

We establish the existence of minimisers in the a priori regularity class \(W^{1,\frac{2p}{p+1}}({\mathbb {D}})\) and show these minimisers have a pseudo-inverse - a continuous \(W^{1,2}({\mathbb {D}})\) surjection of \({\mathbb {D}}\) with \((h\circ f)(z)=z\) almost everywhere. We then give a sufficient condition to ensure \(C^{\infty }({\mathbb {D}})\) smoothness of solutions to the distributional equation. For instance \({\mathbb {K}}(z,f)\in L^{p+1}_{loc}({\mathbb {D}})\) is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further \({\mathbb {K}}(w,h)\in L^1({\mathbb {D}})\) 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.