Abstract

For the Kipriyanov operator \(\Delta _B\) written in the form of the sum of singular differential Bessel operators with, generally speaking, negative parameters, we obtain a representation in spherical coordinates (the Kipriyanov–Beltrami operator). The operator \(\Delta _B\) on the sphere and the corresponding spherical functions (\(B \)-harmonics) are introduced. The main properties of the operator \( \Delta _B\) on the sphere and the differential equation of \(B \)-harmonics are provided. A solution of the inner singular Dirichlet problem in a ball centered at the origin in \(\mathbb {R}^n \) is given. The solution is obtained by the Fourier method in the form of Laplace series in \(B\)-harmonics. The solution is bounded only in the case where all parameters of the Bessel operators occurring in \(\Delta _B \) belong to the interval \((-1,2/n-1) \), \(n\in \mathbb {N}\), \(n\geq 3 \).

中文翻译：

---

具有贝塞尔算子负维的Kipriyanov–Beltrami算子和$$ B $$-调和方程的奇异Dirichlet问题

摘要

对于以奇异差分Bessel运算符之和形式写成的Kipriyanov运算符\（\ Delta _B \），通常来说，它们是负参数，我们得到了球坐标系的表示形式（Kipriyanov–Beltrami运算符）。介绍了球上的算符\（\ Delta _B \）和相应的球面函数（\（B \）- harmonics）。提供了球上算子\（\ Delta _B \）的主要性质和\（B \）-谐波的微分方程。以\（\ mathbb {R} ^ n \）为原点的球中的奇异Dirichlet问题的解给出。通过傅立叶方法以\（B \）-调和的拉普拉斯级数形式获得解。仅在\（\ Delta _B \）中出现的Bessel运算符的所有参数都属于区间\（（-1,2 / n-1）\），\（n \ in \ mathbb中的情况下，该解才有界 {N} \）， \（n \ geq 3 \）。