Comparing Harmonic and Inframonogenic Functions in Clifford Analysis

Boundary value problems for a second‐order elliptic partial differential equation system in Euclidean space

“…){\partial}_{\underset{\_}{x}} $$ keeps the space of $$$ k $$$‐vector fields invariant, and it is verified that $$$$ {\partial}_{\underset{\_}{x}}{F}_k{\partial}_{\underset{\_}{x}}={\left(-1\right)}^k\left({\partial}_{\underset{\_}{x}}\cdotp {\partial}_{\underset{\_}{x}}\wedge {F}_k-{\partial}_{\underset{\_}{x}}\wedge {\partial}_{\underset{\_}{x}}\cdotp {F}_k\right). $$$$ For a recent summary and overview on the inframonogenic function theory, we refer the reader to [7, 8, 11, 14, 15].…”

“…For a recent summary and overview on the inframonogenic function theory, we refer the reader to [7,8,11,14,15]. , called the Moisil-Théodoresco operator; see [18].…”

“…The solutions of this "sandwich" equation were called inframonogenic functions. Given its diverse characteristics, the subject of inframonogenicity has been of increasing interest for mathematicians [7][8][9][10][11][12][13][14][15]. In particular such a functions have been applied to the well-known Lamé-Navier system, which is of considerable interest in practice [16,17].…”

“…The existence of the representation, when the boundary condition ( 14) is fixed, follows from the fact that the Neumann problem associated to (14):…”

Reduced‐quaternion inframonogenic functions on the ball

“…In earlier studies [1, 2], the term “inframonogenic function” was used for solutions of a two‐sided or “sandwich” second‐order differential equation of the form $$$$ \overline{\partial}f\overline{\partial}=0. $$$$ In the papers cited as well as in more recent work [3–7], the symbol $$$ \overline{\partial} $$$ refers to the Dirac operator $$$ {\sum}_{i=1}^n{\partial}_i{e}_i $$$, where $$$ {\partial}_i $$$ denotes $$$ \partial /\partial {x}_i $$$, while $$$ {e}_i $$$ are the units of the Clifford algebra under consideration. A rather different theory results when one uses for $$$ \overline{\partial} $$$ what is sometimes called the generalized Cauchy–Riemann (or Fueter) operator $$$ {\partial}_0+{\sum}_{i=1}^n{\partial}_i{e}_i $$$, as we propose to carry out here.…”