Kipriyanov–Beltrami Operator with Negative Dimension of the Bessel Operators and the Singular Dirichlet Problem for the $$B $$-Harmonic Equation

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 $$(-1,2/n-1) $$ , $$n\in \mathbb {N}$$ , $$n\geq 3 $$ .