Fundamental solutions for second order elliptic operators in Clifford-type algebras
The authors are researching a Laplace type of operator connected to Clifford type algebras. An algebra is of Clifford type if it is generated by elements \(e_{j}\) satisfying the rules \(e_{j}^{2}=-\alpha _{j}\) and \( e_{j}e_{i}+e_{j}e_{i}=2\gamma _{ij}\) for some real parameters \(\alpha _{j}\) and \(\gamma _{ij}=\gamma _{ji}\) . These algebras were introduced by \textit{W. Tutschke} and \textit{C. J. Vanegas} [Some topics on value distribution and differentiability in complex and \(p\)-adic analysis. Beijing: Science Press. 430--450 (2008; Zbl 1232.30034)]. In these algebras, the Cauchy Riemann operator and monogenic functions may be defined formally in the same way as in usual Clifford algebras. Monogenic functions satisfy the second-order differential equation \[ \widetilde{\triangle }u=\overline{D}Du=\partial _{0}^{2}u+\sum_{j=1}^{n}\alpha _{j}\partial _{j}^{2}u-2\sum_{i<j}\gamma _{ij}\partial _{i}\partial _{j}u=0. \] In this paper, the authors find the fundamental solution for this equation and an integral representation formula. The fundamental solution is a kind of generalized Newtonian potential. Also, an integral representation of the operator \(\widetilde{\triangle }-\lambda \) and higher-order \(\widetilde{\triangle}\) operators are studied.
scientific article; zbMATH DE number 6484652
scientific article