This work deals with the boundary value problems for non-regular partial differential equations in the half-plane. Efficient methods are developed for resolution of boundary value problems for improperly elliptic equations - based on the theory of analytic functions and functional analysis - having great theoretical and practical importance. In canonical domains (circle, ellipse, half-plane, etc) explicit formulas for solutions to Riemann-Hilbert and Dirichlet type problems for improperly elliptic equations are obtained. A new approach is proposed for the investigation of the harmonic oscillations of electromagnetic fields in stratified and non-homogeneous media and for the calculation of capacitors bounded by analytic surfaces.
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