Comparing harmonic and inframonogenic functions in Clifford analysis
Let \(\mathbb{R}_{0, m}\) be the real Clifford algebra generated by \(e_1, e_2, \dots, e_m\) over the field of real numbers \(\mathbb{R}\). Consider \(\Omega\) to be a domain of \(\mathbb{R}^m\), a \(\mathbb{R}_{0, m}\)-valued function \( f \in C^2(\Omega)\) is called inframonogenic function if \(\partial_{\underline{x}} f \partial_{\underline{x}} =0\), where \(\partial_{\underline{x}} = \sum_{k=1}^m \partial_{x_k} e_k\) is the Dirac operator. Introducing the operator \(\Psi : \mathbb{R}_{0, m} \to \mathbb{R}_{0, m} \) defined by \(\Psi(f) = \sum_{k=1}^m e_ k f e_k\), where \(f : \mathbb{R}^{ m}\to \mathbb{R}_{0, m}\), it is proved that, for the case \(m\) is odd, a function \(f\) is inframonogenic (harmonic) in \(\Omega \subset \mathbb{R}^m\) if and only if \(\Psi(f)\) is inframonogenic (harmonic) in \(\Omega\). Furthermore, the mean value property for inframonogenic functions is introduced, for which, in contrast to the case of harmonic functions, the average needs to involve on the bounday of the ball of center at the origin and radius \(r>0\) (\(\partial B(0, r)\)) the values of the function as well as those of its first-order partial derivatives. The maximum principle for this class of functions is also discussed. The uniqueness and existence of solution, for \(m\) odd, of the Dirichlet problem \[ \left\{ \begin{array}{lcll} \partial_{\underline{x}} u \partial_{\underline{x}}& =& 0, &\text{ in }\; B(0, r)\\ u &=& g,&\text{ in }\; \partial B(0, r) \end{array}\right. \] assuming \(g\) to be a polynomial, is presented. This solution depends on a harmonic polynomial \(\mathcal{H}\) and the inverse operator of \(\Psi\) applied to \(\partial_{\underline{x}} \mathcal{H} \partial_{\underline{x}}\).
scientific article; zbMATH DE number 7465524
scientific article