Harmonic Green functions for the Almaty apple

Abstract

The parqueting reflection principle is suitable for constructing harmonic Green functions for certain plane domains with boundaries consisting of sub-arcs from circles and straight lines. The (continued) reflection at the boundary parts has to provide a parqueting of the complex plane. Occasionally, however, a parqueting of is not sufficient and multiple parqueting is required, e.g. for the disc sector with opening angle , where a double parqueting is needed, then compensated by applying the square root mapping. The parqueting-reflection principle is more convenient to apply than basing on the conformal invariance of the harmonic Green functions, see e.g., but sometimes the latter is applicable where the first is not. Here the situation is studied for the Almaty apple, a modified half disc. While the bigger of this cut disc is also requiring a double parqueting, for the smaller part a simple one suffices. However, for the bigger part a proper square root reduction to a simple parqueting is still to be determined in order to find the harmonic Green (and Neumann) function. The idea to use this disc cap for a parqueting of the complex plane has occured while two of the authors were visiting a monument of an apple in Almaty, the centre of an area full of wild apple trees considered as the locus of origin of the apple fruit.