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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.

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
Full Text: DOI
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