Задачи типа Гильберта для уравнения Коши–Римана с сингулярной окружностью в младших коэффициентах

Abstract

The Cauchy-Riemann system of equations occupies a special place in the class of first-order elliptic systems. Such a system with lowest terms and a right-hand side is called the generalized Cauchy-Riemann system (GCRS), which can be conveniently studied by making a transition from a real space into a complex space. There are several different mathematical theories of equations that generalize the methods of the theory of functions of a complex variable. In this regard, the work of L. Bers, in which the integration operations with respect to a complex variable for a generalized system of Cauchy-Riemann type are generalized, should primarily be mentioned. This approach has received a well-known finalization in the theory of pseudo-analytic functions. In the works of G.N. Polozhiy, the theory of p-analytic functions was developed, which is close in its ideas to the works of L. Bers. Another, more progressive research area, called «generalized analytical functions», was developed by the school of I.N. Vekua and his followers (B.V. Boyarsky and others). Here, the idea of correspondence between the functions of a complex variable and the solutions of the generalized Cauchy–Riemann equation using the functional analysis techniques is developed. The Vekua theory is constructed on the assumption that the coefficients of the function’s lowest terms belong to the space of summable functions with degree р > 2. The coefficients of such systems may admit «weak» singularities limited by the requirement of p-integrability. Thus, the Vekua theory does not cover even equations with coefficients having first-order singularities. However, other problems, the lowest coefficients of which admit first- order singularities or «strong singularities», boil down to GCRS. In this study, Hilbert type problems are solved for the GCRS the lowest coefficients of which admit strong singularity in a circle. &nbsp