Periodic Boundary Value Problem for a Linear Elliptic Equation of the Second Order in a Half-Plane

We study periodic boundary value problems for a second-order linear partial differential equation with constant coefficients in a half-plane under various assumptions about the roots of the characteristic equation. If the imaginary parts of these roots are of the same sign, then we consider a boundary value problem of the type of the Hilbert problem, and if they are of opposite signs, then we consider a problem of the Dirichlet type. Necessary and sufficient solvability conditions are obtained, explicit formulas are given for the solutions of these problems, and the number of solutions of the corresponding homogeneous problems is found.