Chapter 7 - The objects concept: Harmonic and polyharmonic functions in annuli in ℝ2

Abstract

This chapter discusses harmonic and polyharmonic functions in annuli in R2. The most important fact about the functions harmonic in domains and in particular the annulus is the solubility of the Dirichlet problem. It plays a major role in understanding the structure of the space of harmonic and polyharmonic functions. The annulus plays an important role in the space of polyharmonic and biharmonic functions. Evidently, every harmonic function is also biharmonic; however, not every biharmonic function is harmonic always. The space of all continuous harmonic functions in the annulus and the ball is “parametrized” through the boundary value function because of the solubility of the Dirichlet problem. The Dirichlet problem for the polyharmonic equation provides a “natural parametrization” of the space of polyharmonic functions of order p. However, there is also another natural way to parametrize the space of polyharmonic functions of order p. It is provided by the Almansi formula. This chapter discusses concepts related to Harmonic and polyharmonic functions in the annulus and circle. The “parametrization” of the space of harmonic and polyharmonic functions in the annulus and the ball is also discussed in the chapter.