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

We study minimisers of the p-conformal energy functionals, \(\begin{aligned} \textsf{E}_p(f):=\int _{\mathbb {D}}{\mathbb {K}}^p(z,f)\,\text {d}z,\quad f|_{\mathbb {S}}=f_0|_{\mathbb {S}}, \end{aligned}\) $$\begin{array}{c}\hfill {\mathsf{E}}_p(f):={\int }_{\mathbb{D}}{\mathbb{K}}^p(z,f)\text{d}z,f{{|}_{\mathbb{S}}=f_0|}_{\mathbb{S}},\end{array}$$ defined for self mappings \(f:{\mathbb {D}}\rightarrow {\mathbb {D}}\) $f:\mathbb{D}\to \mathbb{D}$ with finite distortion and prescribed boundary values \(f_0\) $f_0$ . Here \(\begin{aligned} {\mathbb {K}}(z,f) = \frac{\Vert Df(z)\Vert ^2}{J(z,f)} = \frac{1+|\mu _f(z)|^2}{1-|\mu _f(z)|^2} \end{aligned}\) $$\begin{array}{c}\hfill \mathbb{K}(z,f)=\frac{{\Vert Df(z)\Vert }^2}{J(z,f)}=\frac{1+|{\mu }_f{(z)|}^2}{1-|{\mu }_f{(z)|}^2}\end{array}$$ is the pointwise distortion functional and \(\mu _f(z)\) ${\mu }_f(z)$ is the Beltrami coefficient of f. We show that for quasisymmetric boundary data the limiting regimes \(p\rightarrow \infty \) $p\to \infty$ recover the classical Teichmüller theory of extremal quasiconformal mappings (in part a result of Ahlfors), and for \(p\rightarrow 1\) $p\to 1$ recovers the harmonic mapping theory. Critical points of \(\textsf{E}_p\) ${\mathsf{E}}_p$ always satisfy the inner-variational distributional equation \(\begin{aligned} 2p\int _{\mathbb {D}}{\mathbb {K}}^p\;\frac{\overline{\mu _f}}{1+|\mu _f|^2} \varphi _{\overline{z}}\; \text {d}z=\int _{\mathbb {D}}{\mathbb {K}}^p \; \varphi _z\; \text {d}z, \quad \forall \varphi \in C_0^\infty ({\mathbb {D}}). \end{aligned}\) $$\begin{array}{c}\hfill 2p{\int }_{\mathbb{D}}{\mathbb{K}}^p\frac{\overline{{\mu }_f}}{1+|{\mu }_f{|}^2}{\phi }_{\overline{z}}\text{d}z={\int }_{\mathbb{D}}{\mathbb{K}}^p{\phi }_z\text{d}z,\forall \phi \in C_0^{\infty }(\mathbb{D}).\end{array}$$ We establish the existence of minimisers in the a priori regularity class \(W^{1,\frac{2p}{p+1}}({\mathbb {D}})\) $W^{1,\frac{2p}{p+1}}(\mathbb{D})$ and show these minimisers have a pseudo-inverse - a continuous \(W^{1,2}({\mathbb {D}})\) $W^{1,2}(\mathbb{D})$ surjection of \({\mathbb {D}}\) $\mathbb{D}$ with \((h\circ f)(z)=z\) $(h\circ f)(z)=z$ almost everywhere. We then give a sufficient condition to ensure \(C^{\infty }({\mathbb {D}})\) $C^{\infty }(\mathbb{D})$ smoothness of solutions to the distributional equation. For instance \({\mathbb {K}}(z,f)\in L^{p+1}_{loc}({\mathbb {D}})\) $\mathbb{K}(z,f)\in L_{\mathit{loc}}^{p+1}(\mathbb{D})$ is enough to imply the solutions to the distributional equation are local diffeomorphisms. Further \({\mathbb {K}}(w,h)\in L^1({\mathbb {D}})\) $\mathbb{K}(w,h)\in L^1(\mathbb{D})$ 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.