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Harmonic Green functions for the Almaty apple. (English) Zbl 1467.35113

This article describes how harmonic Green functions can be created by using the parqueting-reflection principle. A set of domains \(\{D_j\}\) in the complex plane is called a parqueting of \(\mathbb{C}\) if the sets \(D_j\) are mutually disjoint and the total union of their closure is the whole complex plane \(\mathbb{C}\). A domain \(D\) of the complex plane with piecewise smooth boundary is called admissible for the parqueting-reflection, if continued reflections at the boundary parts achieve a parqueting of \(\mathbb{C}\) with possible exceptions of singular points. The authors discuss theparqueting-reflection principle for the Almaty apple \(D=\{|z|<1:z+\bar z<1\}\) and its complement with respect to the unit disc.

MSC:

35J08 Green’s functions for elliptic equations
35A08 Fundamental solutions to PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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