Boundary value problems on a certain acorn (Q6656812)

An acorn is a circular triangle with three right angles. The advantage of this fact is that continued reflections at the boundary arcs are providing a finite covering of the complex plane by just 8 circular domains. Such a finite parqueting domain is a proper candidate for the parqueting-reflection principle. It serves, in principal, to construct the analytic Schwarz kernel function and the harmonic Green and Neumann functions for the domains from the finite parqueting. In this article the Schwarz, the Dirichlet and the Neumann boundary value problems are explicitly solved for the Cauchy-Riemann equation and solvablity conditions for the latter two problems are given. However, the Neumann problem is just treated for analytic functions. It is based on the Dirichlet problem, which itself is handled as the Schwarz problem by applying the Cauchy-Pompeiu representation properly. Some attention is paid to the behaviour of the solutions in the corner points.