Three boundary value problems for complex partial differential equations in the lens domain (Q6588160)

Parqueting-reflection is used to treat the Schwarz and the Dirichlet problem for Cauchy-Riemann equation, and the Neumann problem for analytic functions. Although the covering of the plane is not explained in detail, it follows from geometrical facts. The Schwarz problem is worked out clearly, while the other two problems are based on Theorem 4.1, presenting a representation formula based only on two of the four reflection points involved in the parqueting process. Therefore the verification of the boundary behavior of the solutions on the boundary of the lens will be doubtful. By the way, the auxiliary function \(f(z)\) in the proof of the Neumann problem is anti-analytic and in providing a proof for the Schwarz problem also the verification of the side condition is required.