On the uniqueness of extremal mappings of finite distortion

Abstract

For an arbitrary convex function $\Psi :[1,\infty )\to [1,\infty )$, we consider uniqueness in the following two related extremal problems: Problem A boundary value problem: Establish the existence of, and describe the mapping $f$, achieving \[ \inf_f \Big\{ \int_{\Bbb D} Ψ({\Bbb K}(z,f))\; dz : f:\bar{\Bbb D} \to \bar{\Bbb D} \; \mbox{a homeomorphism in $W_0^{1,1}(\mathbb{D})+f_0$} \Big\}. \] Here the data $f_0:\bar{\mathbb{D}}\to \bar{\mathbb{D}}$ is a homeomorphism of finite distortion with ${\int }_{\mathbb{D}}\Psi (\mathbb{K}(z,f_0))dz<\infty$ -- a barrier. Next, given two homeomorphic Riemann surfaces $R$ and $S$ and data $f_0:R\to S$ a diffeomorphism. \noindent{\bf Problem B} {\em (extremal in homotopy class):} Establish the existence of, and describe the mapping $f$, achieving \[ \inf_f \Big\{ \int_R Ψ({\Bbb K}(z,f))\; \;dσ(z) : \mbox{$f$ a homeomorphism homotopic to $f_0$} \Big\}. \] There are two basic obstructions to existence and regularity. These are first, the existence of an Ahlfors-Hopf differential and second that the minimiser is a homeomorphism. When these restrictions are met (as they often can be) we show uniqueness is assured. These results are established through a generalisation the classical Reich-Strebel inequalities to this variational setting.