SOME BOUNDARY VALUE PROBLEMS FOR THE INHOMOGENEOUS CAUCHY-RIEMANN EQUATION IN INTERSECTION OF TWO CIRCLES WITH SAME RADIUS

Abstract

This article examines boundary value problems for the Cauchy–Riemann equations in the intersection region of two circles of equal radius. Using the parquet reflection method and the Cauchy–Pompeiu integral formula, a Schwarz representation formula is derived. The parquet reflection method is applied to solving a number of boundary value problems for specific domains whose boundaries consist of circular arcs. This contributes to the method for solving the Schwarz boundary value problem for the nonhomogeneous Cauchy-Riemann equation and also plays an important role in solving other boundary value problems. If the boundary of a domain has the specified properties, a new domain can be created in the plane by reflecting the original domain about its boundary, and by reflecting this new domain about its boundary, another domain is found. By repeating this process, we obtain fragments of the plane, the union of which provides a covering for the complex plane. The covering can be achieved with a single reflection, several successive reflections, or infinitely repeated reflections. Furthermore, the Schwarz problem for the inhomogeneous Cauchy–Riemann equation is studied; explicit solution formulas are obtained, and solvability conditions for the boundary value problem under consideration are established.