On the structure of the kernel of the Schwarz problem in an ellipse in the general case

The paper calculates the structure of the kernel and co-kernel of the Schwartz problem for J-analytic functions defined in the ellipse D with a boundary Γ. The Schwartz problem consists in finding a J-analytic function in the ellipse D by the known value of its real part on Γ. In paragraphs 1 and 2 the problem is formulated and its solution for a special right part is studied. Paragraph 3 contains the necessary information from one paper by A.P. Soldatov. Paragraph 4 constructs the solution of the Schwarz union problem for the special right-hand side. On the basis of these results, paragraph 5 calculates the kernel and the co-kernel of the Schwartz problem. The model of their calculation is briefly described at the beginning of the fifth paragraph. Then in the theorems 5.1–5.6 this scheme is implemented. Here the notions of theoretical and algorithmic solvability of the special Schwarz problem introduced by the author are used. The method of mathematical induction is used as well. It is shown that the kernel and co-kernel of the Schwarz problem in an ellipse consist only of vector polynomials. The paper describes the structure of the kernel and co-kernel in terms of the ranks of some real matrices depending on the matrix J and the ellipse Γ. The paper concludes with an example of calculating the kernel of the Schwarz problem in an ellipse for a two-dimensional matrix J with multiple eigenvalue.