

Seminar on Complex Analysis (Gonchar Seminar)
June 8, 2015 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)






The singular Riemann–Hilbert problem in compexshaped domains and some applications
V. I. Vlasov^{}, S. I. Bezrodnykh^{} ^{} Dorodnitsyn Computing Centre of the Russian Academy of Sciences, Moscow

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Abstract:
In simply connected domains $\mathscr{B}$ of complex shape, the Riemann–Hilbert problem with
discontinuous data and growth of solution at some points of the boundary is considered. By the use of conformal mapping $\mathscr{B}$ onto $\mathbb{H}^+$, this problem is reduced to an analogous one in $\mathbb{H}^+$.
A method for solving the latter problem is given in terms of modified Cauchytype integral. In the case of piecewise constant data of the problem, a fundamentally new representation of desired analytic function is obtained in the form of Christoffel–Schwarztype integral, which solves the problem posed by Riemann of geometric interpretation of the solution. This form of solution is more convenient for numerical implementation than the conventional
representation in terms of Cauchytype integrals. Examples of application of the obtained results to some physical problems with numerical realization are given.

