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Comparing Harmonic and Inframonogenic Functions in Clifford Analysis

Abstract:

Harmonic functions are the solutions of the second-order partial differential equation ∂x̲∂x̲u=0, where ∂x̲ stands for the Dirac operator factorizing the Laplacian in Rm. In this paper, we consider functions satisfying the sandwich equation ∂x̲u∂x̲=0, the so-called inframonogenic functions. It is easily seen that the real-valued solutions of both previous equations will be identical. However, the situation is quite different when Clifford algebra valued solutions are considered. This leads to different classes of functions, which appear together in some topics of linear elasticity theory. The main purpose of this paper is to deepen the understanding of inframonogenic functions as well as to contrast its behavior with the more traditional harmonic functions.