Approximation in the Cone of Positive Harmonic Functions

Abstract

In the course of studying quadrature domains Gustafsson, Sakai and Shapiro were led to the question of whether it is the case that the positive integrable harmonic functions on a bounded domain are dense among all positive harmonic functions (w.r.t. uniform convergence on compact subsets). In this article we will show how such approximation problems are related to representing measures on the Martin boundary, and then we use these results to give a counterexample to the question posed above.