This book deals with the boundary value problems 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. The book can be used by professionals in a wide range of specialities, including mathematics, mechanics, physics and many realms of engineering.