有限偏差映射的加权Grtzsch问题  

Weighted Grtzsch Problem for Finite Distortion Mappings

作　　者：冯小高[1,2] 吴冲[3] 唐树安[4] FENG Xiaogao WU Chong TANG Shu＇an(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 School of Mathematical Sciences, Southwest Jiaotong University, Chengdu 611756, China School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001,China.)

机构地区：[1]苏州大学数学科学学院,江苏苏州215006 [2]西华师范大学数学与信息学院,四川南充637002 [3]西南交通大学数学学院,成都611756 [4]贵州师范大学数学科学学院,贵阳550001

出　　处：《数学年刊（A辑）》2016年第4期359-366,共8页Chinese Annals of Mathematics

基　　金：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);西华师范大学科研启动资助项目(No.13D017)的资助

摘　　要：考虑如下的极值问题：inf f∈F ∫∫Q1 φ（K（z,f））λ（x）｜dz｜2,其中F是从矩形Q1到矩形Q2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，φ是正的严格凸的递增函数，而λ（x）是正的加权函数．作者在文“Sci China Math，2016，59（4）：673-686”中证明了当φ′无界时，上述极值问题存在唯一的极值映射f0）z）=u（x）＋iy．本文考虑φ′有界的情形，得到如下结果：当L〈l时，上述极值问题也存在唯一的极值映射；但当L〉l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∈F ∫∫Q1 φ（K（z,f））λ（x）｜dz｜2 where F denotes the set of all homeomorphims f with finite linear distortion K （z, f） between two rectangles Q1 and Q2 taking vertices into vertices, φ is a strictly convex increasing positive function and ）λ（x） is a positive weighted function. In ＂Sci China Math, 2016, vol. 59, no. 4, pp. 673-686＂, the authors proved that when φ′ is unbounded the extremal problem exists uniquely an extremal mapping with the form of fo（z） = u（x）＋iy. In this paper, the authors consider the case that φ′ is bounded. It is obtained that when L 〈 l,there also exists uniquely an extremal mapping, while, when L 〉 l, 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.

关 键 词：Grotzsch问题 有限偏差映射 极值映射 

分 类 号：O174.55[理学—数学]