Weighted Gr$\ddot{\textbf{o}}$tzsch Problem for\\ Finite Distortion Mappings
Received: June 12, 2015  Revised: December 14, 2015
DOI：

 Author Name Affiliation FENG Xiaogao Corresponding author. \!\!School of Mathematical Sciences, Soochow University, Suzhou 215006, Jiangsu, China; College of Mathematics and Information, China West Normal University, Nanchong 637002, Sichuan, China. WU Chong School of Mathematical Sciences, Southwest Jiaotong University, Chengdu 611756, China. TANG Shu'an School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China.
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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.