Harmonic balls and the two-phase Schwarz function

• In this article, we introduce the concept of harmonic balls in sub-domains of 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.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

Nyckelord

Schwarz function

mean-value property

harmonic functions

two-phase free boundary

quadrature domains

Primary: 35R35

31A05

31B05

31B20

MATHEMATICS

MATEMATIK

Publikations- och innehållstyp

ref (ämneskategori)

art (ämneskategori)