论文标题
关于有限失真的极端映射的独特性
On the uniqueness of extremal mappings of finite distortion
论文作者
论文摘要
对于任意凸函数$ψ：[1，\ infty）\ to [1，\ infty）$，我们在以下两个相关的极端问题中考虑唯一性： 问题一个边界值问题：建立并描述映射$ f $的存在，实现\ [\ inf_f \ big \ \ {\ int _ {\ bbb d}ψ（{\ bbb k}（z，z，f））\; dz：f：\ bar {\ bbb d} \ to \ bar {\ bbb d} \; \ mbox {$ w^{1,1} _ {0}（{\ bbb d}）+f_0 $} \ big \}中的同构。 \]这里的数据$ f_0：\ bar {\ bbb d} \ to \ bar {\ bbb d} $是具有有限失真的同态，$ \ int _ {\ bbb d}ψ（{\ bb k}（\ bb k}（z，f_0））\; DZ <\ infty $ - 障碍。接下来，给定两个同构riemann表面$ r $和$ s $和data $ f_0：r \ to s $ diffemormormormism。 \ noindent {\ bf问题b} {\ em（同型类中极端）：}建立并描述映射$ f $的存在，实现\ [\ inf_f \ big \ big \ {\ int_r {\int_rψ（{\ bb k}（\ bb k}（z，z，f）（z，f））\; \;dσ（z）：\ mbox {$ f $ a同态构态同型至$ f_0 $} \ big \}。 \] 存在两个基本障碍和规律性。这些首先是Ahlfors-Hopf差异的存在，其次是最小化器是同态性。当满足这些限制时（通常可以），我们会确保独特性。这些结果是通过概括来确定的，经典的帝国帝国strebel不平等现象。
For an arbitrary convex function $Ψ:[1,\infty) \to [1,\infty)$, 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 \[ \inf_f \Big\{ \int_{\Bbb D} Ψ({\Bbb K}(z,f))\; dz : f:\bar{\Bbb D} \to \bar{\Bbb D} \; \mbox{a homeomorphism in $W^{1,1}_{0}({\Bbb D})+f_0$} \Big\}. \] Here the data $f_0:\bar{\Bbb D} \to \bar{\Bbb D}$ is a homeomorphism of finite distortion with $\int_{\Bbb D} Ψ({\Bbb K}(z,f_0))\; dz<\infty$ -- 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 \[ \inf_f \Big\{ \int_R Ψ({\Bbb K}(z,f))\; \;dσ(z) : \mbox{$f$ a homeomorphism homotopic to $f_0$} \Big\}. \] 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.
