The singular Riemann–Hilbert problem in compex-shaped domains and some applications

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 Cauchy-type 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–Schwarz-type 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 Cauchy-type integrals. Examples of application of the obtained results to some physical problems with numerical realization are given.