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$$\mathbb R$$-linear problem for multiply connected domains and alternating method of Schwarz. (English. Russian original) Zbl 1276.30052
J. Math. Sci., New York 189, No. 1, 68-77 (2013); translation from Sovrem. Mat. Prilozh. 77 (2012).
The authors apply a version of the alternating method of Schwarz to a system of boundary value problems involving several disjoint domains $$D_k$$ in the complex plane $$\mathbb C$$. Suppose that each $$D_k$$ is bounded by a smooth simple curve $$L_k$$, oriented counterclockwise. Let $$a,b,c$$ be Hölder continuous functions on $$L=\bigcup L_k$$ such that $$a\neq 0$$. Also denote $$D=\overline{\mathbb C}\setminus \bigcup \overline{D_k}$$. The $$\mathbb R$$-linear conjugation problem is to find a function $$\varphi$$ that is holomorphic in the complement of $$L$$, is continuous up to the boundary of each domain $$D, D_k$$, and satisfies the equation $$\varphi^+=a\varphi^{-} + b \overline{\varphi^{-}} +c$$ on $$L$$. Here $$\varphi^{+}$$ and $$\varphi^{-}$$ stand for the boundary values of $$\varphi$$ on $$L$$ as approached from $$D$$ and from $$D_k$$, respectively. In addition, $$\varphi$$ must vanish at infinity.
The following result (Theorem 1 in the paper) generalizes a 1960 theorem by the first-named author. Suppose in addition that $$|b|<a$$ on $$L$$, and let $$\kappa$$ be the winding number of $$a$$ on $$\partial D$$. If $$\kappa\geq 0$$, then the problem stated above has a solution. Furthermore, the homogeneous problem ($$c=0$$) has $$2\kappa$$ $$\mathbb R$$-linearly independent solutions vanishing at infinity. If $$\kappa<0$$, then the solution does not necessarily exist.
The article includes a historical overview of this and related boundary value problems, supported by a long list of references.

##### MSC:
 30E25 Boundary value problems in the complex plane 30D05 Functional equations in the complex plane, iteration and composition of analytic functions of one complex variable
##### Keywords:
conjugation problem; alternating method; integral equations
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