| 冯小高,吴 冲,唐树安.有限偏差映射的加权Grotzsch问题[J].数学年刊A辑,2016,37(4):359~366 |
| 有限偏差映射的加权Grotzsch问题 |
| Weighted Gr$\ddot{\textbf{o}}$tzsch Problem for\\ Finite Distortion Mappings |
| 投稿时间：2015-06-12 修订日期：2015-12-14 |
| DOI： |
| 中文关键词: Gr\"{o}tzsch问题, 有限偏差映射, 极值映射 |
| 英文关键词:Gr\"{o}tzsch problem, Finite distortion mapping, Extremal mapping |
| 基金项目:本文受到国家自然科学基金 (No.11601100, No.11226097),
中央高校基本科研业务费专项资金(No.2682015CX057),
贵州师范大学博士启动基金(No.11904-05032130006)和西华师范大学科研启动资助项目(No.13D017)的资助. |
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| 中文摘要: |
| 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. |
| 英文摘要: |
| 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. |
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