Unions of John domains and uniform domains in real normed vector spaces

Abstract

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-