Boundary Value Problems on a Certain Acorn

For a certain domain with boundaries composed by arcs from three circles, the parqueting-reflection principle is used to reflect points within the domain onto the entire complex plane. By using the Cauchy–Pompeiu formula, we construct the Cauchy–Schwarz representation formula and obtain an explicit solution to the Schwarz boundary value problem for the non-homogeneous Cauchy–Riemann equation. Further, we consider the Dirichlet and Neumann boundary value problems for the Cauchy–Riemann equation, and expressions of solution and the condition of solvability are explicitly obtained.