A Riemann–Hilbert Type Problem for the Generalized Cauchy–Riemann Equation with Supersingular Points on a Half-Plane

Abstract

For an equation with the Cauchy–Riemann operator having strong isolated point singularities in the lowest coefficient, an integral representation of the solution is found in the class of bounded functions at infinity, and a Riemann–Hilbert type problem on a half-plane is investigated. Specialists are well aware about the important role played in applications by I.N. Vekua’s theory of generalized analytic functions of an equation in the case when the coefficients and the right-hand side of the generalized Cauchy–Riemann system belong to the class of summable functions. It is deeply linked with many areas of analysis, geometry and mechanics, including quasi-conformal maps, surface theory, shell theory, and gas dynamics. In particular, it is widely used in modeling transonic gas flows, states of momentless stressed equilibrium of convex shells, and many other processes. The Vekua–Pompeiu integral plays a key role in this theory. When the lowest coefficient has strong singularities, some researchers believe that the Vekua–Pompeiu integral, which is main research engine in the theory of generalized analytic functions, cannot be applied. By using the Vekua–Pompeiu integral, we have succeeded in constructing solutions of the Cauchy–Riemann system the lowest coefficient of which contains strong isolated point singularities in an infinite domain. A Riemann–Hilbert type problem on a half-plane is investigated for the equation under consideration by using an explicit integral representation.