Harmonic balls and the two-phase Schwarz function
Henrik Shahgholian, Tomas Sjödin · Complex Variables and Elliptic Equations · 2011
In this article, we introduce the concept of harmonic balls in sub-domains of ℝ n , through a mean-value property for a subclass of harmonic functions on such domains. In the complex plane, and for analytic functions, a similar concept fails to exist due to the fact that analytic functions cannot have prescribed data on the boundary. Nevertheless, a two-phase version of the problem does exist, and gives rise to the generalization of the well-known Schwarz function to the case of a two-phase Schwarz function. Our primary goal is to derive simple properties for these problems, and tease the appetites of experts working on Schwarz function and related topics. Hopefully these two concepts will provoke further study of the topic.