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Potentials for Three-Dimensional Singular Elliptic Equation and Their Application to the Solving a Mixed Problem
Lobachevskii Journal of Mathematics Pub Date : 2020-07-18 , DOI: 10.1134/s1995080220060086
T. G. Ergashev

Abstract

In earlier research, the double- and simple layer potentials have been successfully applied in solving boundary value problems for two-dimensional elliptic equations. Despite the fact that all fundamental solutions of a three-dimensional elliptic equation with one, two and three singular coefficients were known, the potential theory was not constructed in any case of a singularity. Here, in this paper, our goal is to construct a potential theory corresponding to the three-dimensional elliptic equation with one singular coefficient. We used some properties of Gaussian hypergeometric function to prove the limiting theorems, while deriving integral equations concerning the denseness of potentials.


中文翻译:

三维奇异椭圆方程的势及其在求解混合问题中的应用

摘要

在较早的研究中,双层和简单层势已成功地用于解决二维椭圆方程的边值问题。尽管已知具有一,二和三个奇异系数的三维椭圆方程的所有基本解,但在任何奇异情况下均未构建势能理论。在本文中,我们的目标是构建与具有一个奇异系数的三维椭圆方程相对应的势能理论。我们使用高斯超几何函数的一些性质来证明极限定理,同时推导有关势密度的积分方程。
更新日期:2020-07-18
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