Abstract

In a rectangular domain, for the equation of mixed elliptic-hyperbolic type with the Lavrent’ev–Bitsadze operator and two perpendicular lines of type change, we investigate the first boundary value problem. A criterion for the uniqueness of its solution is established. The solution to the problem is constructed as the sum of a series in the biorthogonal system of the corresponding spectral problem for an ordinary differential operator. On the basis of the completeness of the biorthogonal system in the space of square-summable functions, we prove the uniqueness of a solution to the problem. When proving the existence of a solution, a problem of small denominators arises. Therefore, we obtain estimates on the separation of these denominators from zero with the corresponding asymptotics. This allows us to establish the existence of a solution to the problem from the required class.

中文翻译：

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具有两个内部类型变化的混合线方程的Dirichlet问题

摘要

在矩形域中，对于带有Lavrent'ev–Bitsadze算子和两条类型变化的垂直线的混合椭圆-双曲型方程，我们研究了第一个边值问题。建立了其解决方案唯一性的标准。对于普通微分算子，该问题的解决方案被构造为相应光谱问题在双正交系统中的级数之和。在平方和函数空间内双正交系统的完备性的基础上，我们证明了该问题解的唯一性。当证明解决方案的存在时，会出现分母小的问题。因此，我们获得了关于这些分母从零开始的估计以及相应的渐近性。