Combined periodic boundary-value problems and their applications in the theory of elasticity

Abstract The fundamental boundary value problems of analytic function theory are considered on certain systems of contours possessing translational symmetry: the Riemann problem on an oblique lattice of arbitrary contours, the Hubert problem and the mixed problem for a half-plane, the Dirichlet problem for a plane with a periodic system of slits on a line. In contrast to /1/, where the listed problems are solved under the condition of periodicity of their coefficients and free terms, this condition is here imposed only on the coefficients. By applying a discrete Fourier transform and periodicity of the boundary conditions for an elementary cell, the formulated quasiperiodic problems are reduced to periodic problems and are solved in closed form. The results obtained are used to solve (in quadratures) new mixed problems of elasticity theory in translationally symmetric domains with nonperiodic boundary conditions.