zbMATH — the first resource for mathematics

On boundary value problem of Neumann type for hypercomplex function with values in a Clifford algebra. (English) Zbl 0713.35060
Geometry and physics, Proc. 9th Winter Sch., SrnĂ­/Czech. 1989, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 22, 213-226 (1990).
[For the entire collection see Zbl 0699.00032.]
This paper is concerned with the boundary value problem for hypercomplex function with values in a Clifford algebra: \[ {\bar \partial}_ nw=f,\quad x\in K_ n^{(n)};\quad X^{(n)}((\partial /\partial n)w)=0,\quad x\in \partial K_ n^{(n)}, \] where \({\bar \partial}_ n\) is the generalized Cauchy-Riemann operator \({\bar \partial}_ n=e_ 1(\partial /\partial x_ 1)+\sum^{n}_{j=2}e_ j(\partial /\partial x_ j)\), \(\{e_ 2,...,e_ n\}\) is the basis of \({\mathbb{R}}^{0,n-1}\), \(e_ 1\) is the identity of the Clifford algebra \({\mathbb{R}}_{0,n-1}\), \(X^{(n)}\) is a certain projective operator of \({\mathbb{R}}_{0,n-1}\) onto its subalgebra, and \(K_ n^{(n)}\) is the unit ball in \({\mathbb{R}}^ n.\)
In order to solve this problem, firstly the author constructs an explicit representation formula for Neumann function for Laplacian over the unit ball in \({\mathbb{R}}^ n\) (n\(\geq 3)\) by means of the fundamental solution for Laplacian and Gegenbauer polynomials. Then making use of this Neumann function the solutions of above problem are obtained if the necessary and sufficient condition for the solvability is fulfilled.
Reviewer: Zhenyuan Xu

35N05 Overdetermined systems of PDEs with constant coefficients
35J55 Systems of elliptic equations, boundary value problems (MSC2000)
15A66 Clifford algebras, spinors