UNIONS OF JOHN DOMAINS AND UNIFORM DOMAINS IN REAL NORMED VECTOR SPACES
UNIONS OF JOHN DOMAINS AND UNIFORM DOMAINS IN REAL NORMED VECTOR SPACES
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DOI：
10.5186/aasfm.2010.3539
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发表时间：
2010-08
期刊：
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通讯作者：
Yaxiang Li;Xiantao Wang
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文献类型：
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作者：
Yaxiang Li;Xiantao Wang
设E是维数至少为2的真实的赋范向量空间。本文研究了E中两个John整环的并何时是John整环，E中两个一致整环的并何时是一致整环。1.引言和主要结果在本文中，我们总是假设E表示一个真实的赋范向量空间，dimE <$2，D是E中的一个真子域。E中向量z的范数记为jzj，对于E中任意两点z1;z2，它们之间的距离记为jz 1 i z2 j，端点为z1和z2的闭合线段记为（z1;z2）。对于x ~ 2 E，r > 0，设B（x;r）表示E中以x为中心，半径为r的开球.对于真实的数r和s，我们使用记法：r ^ s = minfr;sg。约翰域在欧氏空间R n介绍了约翰（1）在他的工作弹性。这个术语是由于Martio和Sarvas（3）。粗略地说，一个域是一个约翰域，如果有可能从域的一点旅行到另一点而不太靠近边界。准确的定义如下。定义1.1.称D为c-John整环，如果对任意一对点x1;x2 2 D，都有一条可求长弧∞连接它们，其中对所有的x2 ∞，都有'（∞（x1;x））^ '（∞（x2;x））·cd（x），其中c为正常数，∞（xj;x）表示∞的以xj和x（j = 1;2）为端点的闭子弧，'（∞（xj;x））表示∞（xj;x）的弧长。∞称为连接x1和x2的c-John弧。参见（4）给出John域的几个刻画。在约翰·杜的研究中-
Let E be real normed vector spaces with the dimension at least 2. In this paper we study the following questions: When is the union of two John domains in E a John domain and when is the union of two uniform domains in E a uniform domain? 1. Introduction and main results Throughout the paper, we always assume that E denotes a real normed vector space with dimE ‚ 2 and that D is a proper subdomain in E. The norm of a vector z in E is written as jzj, and for any two points z1;z2 in E, the distance between them is denoted by jz1 i z2j, and the closed line segment with endpoints z1 and z2 by (z1;z2). For x 2 E and r > 0, we let B(x;r) denote the open ball in E with center x and radius r. For real numbers r and s, we use the notation: r ^ s = minfr;sg. John domains in Euclidean spaces R n were introduced by John (1) in connection with his work on elasticity. The term is due to Martio and Sarvas (3). Roughly speaking, a domain is a John domain if it is possible to travel from one point of the domain to another without going too close to the boundary. The precise definition is as follows. Definition 1.1. D is called a c-John domain if for every pair of points x1;x2 2 D there is a rectifiable arc ∞ joining them with '(∞(x1;x)) ^ '(∞(x2;x)) • c d(x) for all x 2 ∞, where c is a positive constant, ∞(xj;x) denotes the closed subarc of ∞ with endpoints xj and x (j = 1;2), '(∞(xj;x)) the arclength of ∞(xj;x). ∞ is called a c-John arc joining x1 and x2. See (4) for several characterizations of John domains. In the study of John do-