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 \[ \inf_f \Big\{ \int_{\Bbb D} \Psi({\Bbb K}(z,f))\; dz : f:\bar{\Bbb D} \to \bar{\Bbb D} \; \mbox{a homeomorphism in $W^{1,1}_{0}({\Bbb D})+f_0$} \Big\}. \] Here the data $f_0:\bar{\Bbb D} \to \bar{\Bbb D}$ is a homeomorphism of finite distortion with $\int_{\Bbb D} \Psi({\Bbb K}(z,f_0))\; dz<\infty$ -- 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 \[ \inf_f \Big\{ \int_R \Psi({\Bbb K}(z,f))\; \;d\sigma(z) : \mbox{$f$ a homeomorphism homotopic to $f_0$} \Big\}. \] 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.