Weighted Gr$\ddot{\textbf{o}}$tzsch Problem for\\ Finite Distortion Mappings

DOI：

 作者 单位 冯小高 通讯作者. 苏州大学数学科学学院, 江苏\苏州 215006; 西华师范大学数学与信息学院,四川\南充637002. 吴　冲 西南交通大学数学学院, 成都 611756. 唐树安 贵州师范大学数学科学学院, 贵阳 550001.

考虑如下的极值问题: $$\inf_{f\in \mathcal{F}}\iint_{Q_{1}}\varphi(K(z,f))\lambda(x)|\rmd z|^{2},$$ 其中$\mathcal{F}$ 是从矩形$Q_1$ 到矩形$Q_2$ 并保持端点且具有有限线性偏差 $K(z,f)$的所有同胚映射$f$的集合, $\varphi$ 是正的严格凸的递增函数, 而$\lambda(x)$ 是正的加权函数. 作者在文{\it Sci China Math}, 2016, 59(4):673--686''中证明了当 $\varphi'$ 无界时, 上述极值问题存在唯一的极值映射$f_{0}(z)=u(x)+\rmi y$. 本文考虑$\varphi'$ 有界的情形, 得到如下结果: 当$Ll$ 时, 极值映射可能不存在. 借助于 Martin 和 Jordens 的方法, 构造了一族最小序列使得其极限达到最小值.

This paper deals with the following extremal problem: $$\inf_{f\in \mathcal{F}}\iint_{Q_{1}}\varphi(K(z,f))\lambda(x)|\rmd z|^{2},$$ where $\mathcal{F}$ denotes the set of all homeomorphims $f$ with finite linear distortion $K(z, f)$ between two rectangles $Q_{1}$ and $Q_{2}$ taking vertices into vertices, $\varphi$ is a strictly convex increasing positive function and $\lambda(x)$ is a positive weighted function. In {\it Sci China Math}, 2016, vol. 59, no. 4, pp. 673--686'', the authors proved that when $\varphi'$ is unbounded the extremal problem exists uniquely an extremal mapping with the form of $f_{0}(z)=u(x)+\rmi y$. In this paper, the authors consider the case that $\varphi'$ is bounded. It is obtained that when $Ll$, there is no solution for the minimization problem. By the method of Martin and Jordens, a minimizing sequence which attains the minimization in the limit is constructed.