Abstrakt
A champagne subdomain of a non-empty connected open set U in R^d, d >= 2, is obtained by omitting pairwise disjoint closed balls B(x, r(x)), the bubbles, where x is an element of X and X is an infinite, locally finite set in U. The union A of these balls may be unavoidable, that is, Brownian motion, starting in U \ A and killed when leaving U, may hit A almost surely or, equivalently, A may have harmonic measure 1 for U \ A.
Recent publications by Gardiner and Ghergu (d >= 3) and by Pres (d = 2) give rather sharp answers to the question of how small such a set A may be, when U is the unit ball. In this paper, using a totally different approach, optimal results are obtained, which hold also for arbitrary connected open sets U.