Abstract

We consider a boundary value problem for an elliptic differential equation with analytic coefficients that is degenerate in one of the variables in a rectangle. Using the method of spectral separation of singularities, a solution of this problem is constructed in the form of a Poisson series—a series in the eigenfunctions of the second-order limit linear ordinary differential operator with analytic coefficients. Estimates are obtained for the functions of the fundamental system of solutions and the Green’s functions of the sequence of boundary value problems corresponding to this operator; this enables one to weaken the previously known conditions for the convergence of the series constructed for the solution, including the case of presence of logarithmic singularities.

中文翻译：

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不规则退化椭圆微分算子解析理论中泊松级数的应用

摘要

我们考虑一个椭圆微分方程的边值问题，该方程的解析系数在矩形中的一个变量中是退化的。利用奇异点谱分离的方法，以泊松级数的形式构造了该问题的解——泊松级数是具有解析系数的二阶极限线性常微分算子的本征函数级数。获得基本解系统的函数和对应于该算子的边值问题序列的格林函数的估计值；这使人们能够削弱先前已知的为解决方案构建的级数收敛的条件，包括存在对数奇点的情况。