2012/04/07 by Barton, Ariel, Ward, Lesley A. · 1 citation
#31A15 #60J65 #Complex Variables (math.CV) #FOS: Mathematics #Primary 30C85 #Secondary 30C20
paper · doi:10.48550/arxiv.1204.1635
Let D be a planar domain containing 0. Let hD(r) be the harmonic measure at 0 in D of the part of the boundary of D within distance r of 0. The resulting function hD is called the harmonic measure distribution function of D. In this paper we address the inverse problem by establishing several sets of sufficient conditions on a function f for f to arise as a harmonic measure distribution function. In particular, earlier work of Snipes and Ward shows that for each function f that increases from zero to one, there is a sequence of multiply connected domains Xn such that hXn converges to f pointwise almost everywhere. We show that if f satisfies our sufficient conditions, then f = hD, where D is a subsequential limit of bounded simply connected domains that approximate the domains Xn. Further, the limit domain is unique in a class of suitably symmetric domains. Thus f = hD for a unique symmetric bounded simply connected domain D.