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Generalized Clifford algebras associated to certain partial differential equations. (English) Zbl 1441.30071

Summary: In the classical Clifford analysis the Laplace operator is factorized by the Cauchy-Riemann operator \(\Delta =\overline{D}D\). The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic function taking values in that algebra satisfy the given partial differential equation. For instance, in order to represent biharmonic functions in the theory of plane elasticity, in 1934 L. Sobrero introduced a hypercomplex algebra which is generated by the imaginary \(e\) with the rule \((e^2+1)^2=0\). In this paper we introduce an extension of the idea of L. Sobrero to construct some generalized Clifford algebras in order to cover more partial differential equations in higher dimensions.

MSC:

30G35 Functions of hypercomplex variables and generalized variables
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