Champagne subdomains with unavoidable bubbles

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• 作者：Wolfhard Hansen ; Ivan Netuka
• 关键词：31A15 ; 31B15 ; 60J65
• 刊名：Advances in Mathematics
• 出版年：2013
• 出版时间：10 September, 2013
• 年：2013
• 卷：244
• 期：Complete
• 页码：106-116
• 全文大小：197 K

文摘

A champagne subdomain of a connected open set in , , is obtained by omitting pairwise disjoint closed balls , , the bubbles, where is an infinite, locally finite set in . The union of these balls may be unavoidable, that is, Brownian motion, starting in and killed when leaving? , may hit? almost surely or, equivalently, may have harmonic measure 1 for .

Recent publications by Gardiner and Ghergu () and by Pres () give rather sharp answers to the question of how small such a set may be, when ?is the unit ball.

In this paper, using a totally different approach, optimal results are obtained, which hold also for arbitrary connected open sets? .