The Riemann-Hilbert problem and the R-linear problem for multiply connected domains in a class of doubly periodic functions on the complex plane are reduced to functional equations which are solved in terms of the Poincare series. The results are applied to calculation of the effective conductivity of the composites with many different circular inclusions in the unit cell and to the permeability of arrays of unidirectional cylinders. The final formula for the effective conductivity tensor involves locations of the centers of inclusions, conductivities of constitutes and radii of inclusions in analytical form. Another problem, the steady viscous flow in curvilinear channels is analytically studied by the method of perturbations.