Abstract

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.

中文翻译：

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半平面中二阶线性椭圆方程的周期边值问题

摘要

在关于特征方程根的各种假设下，我们研究了半平面中具有恒定系数的二阶线性偏微分方程的周期边值问题。如果这些根的虚部具有相同的符号，则考虑希尔伯特问题类型的边值问题，如果它们具有相反的符号，则考虑狄利克雷类型的问题。得到了必要和充分的可解性条件，给出了解决这些问题的显式公式，并找到了相应的同类问题的解数。