Singular Riemann–Hilbert problem in complex-shaped domains

S. I. Bezrodnykh   ; V. I. Vlasov

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 54 (2014) no. 12, pp. 1904-1953

Résumé

In simply connected complex-shaped domains $\mathcal{B}$ a Riemann–Hilbert problem with discontinuous data and growth condidions of a solution at some points of the boundary is considered. The desired analytic function $\mathcal{F}(z)$ is represented as the composition of a conformal mapping of $\mathcal{B}$ onto the half-plane $\mathbb{H}^+$ and the solution $\mathcal{P}^+$ of the corresponding Riemann–Hilbert problem in $\mathbb{H}^+$. Methods for finding this mapping are described, and a technique for constructing an analytic function $\mathcal{P}^+$ in $\mathbb{H}^+$ in the terms of a modified Cauchy-type integral. In the case of piecewise constant data of the problem, a fundamentally new representation of $\mathcal{P}^+$ in the form of a Christoffel–Schwarz-type integral is obtained, which solves the Riemann problem of a geometric interpretation of the solution and is more convenient for numerical implementation than the conventional representation in terms of Cauchy-type integrals.

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