Heinrich Begehr

1. Polyharmonicity, as a generalization of harmonicity, has inspired much research a long time ago already [1–5]. Principally, the polyharmonic equation can be viewed as an iterated set of harmonic...

The harmonic Robin function is redefined as an interpolation between the related Green and Neumann functions and is explicitly constructed by conformal mapping for some bounded and an unbounded doubly connected ring domain.

As the Green and the Neumann functions are invariant under inversive transformations this is also true for the harmonic Robin function. This fact is proved here, implying also its conformal invariance. From the Robin function for the unit disc a respective conformal mapping gives the Robin function for the upper half plane. The Schwarz problem for analytic functions provides a direct calculation of the Robin function.

An explicit formula for the polyharmonic Green function of any order for the unit disc of the complex plane based on the work of E. Almansi is widely known through I.N. Vekua’s 1967 book. A simple, however, not easy way to prove the formula is by verification of the defining properties. P.R. Garabedian has suggested a constructive way for this function in his 1964 pde book but only has applied it for the biharmonic case. His construction is not smoothly working for higher order as is explained here in the triharmonic case. The general case will demand very concentrated calculations. The procedure should similarly work also for the half plane.

Convolution of harmonic Green functions leads to different hybrid higher order polyharmonic Green functions. They differ from the related polyharmonic Green–Almansi function and are linked to different boundary value problems. The advantage of convoluted higher order Green functions is that they allow to decompose the boundary value problem for a linear higher order Poisson equation into some for a system of fist order Poisson equations. The higher the order the greater is the variety of possible boundary value problems and related hybrid Green functions. The situation is explained here just for the biharmonic operator. In order to be explicit just the unit disc is treated.

\title {\bf Review on\\ Impossibilities, mathematics, and logic\\ by F. A. Mala} \date{} \maketitle \noindent In fact mathematics is some art and the tool for this art is proving. The author explains some generally known basic facts where proofs m...