Generalizations of harmonic functions in $${\mathbb R}^m$$

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].…”

“…We are now in a position to consider certain classes of second‐order partial differential equations based on two given structural sets $$$ \nu $$$ and $$$ \vartheta $$$, that is, the general equation $$$$ {}^{\nu }{\partial}_{\underset{\_}{x}}{f}^{\vartheta }{\partial}_{\underset{\_}{x}}=0 $$$$ and $$$$ {}^{\nu }{\partial_{\underset{\_}{x}}}^{\vartheta }{\partial}_{\underset{\_}{x}}f=0, $$$$ whose solutions have been referred to as $$$ \left(\nu, \vartheta \right) $$$‐inframonogenic and $$$ \left(\nu, \vartheta \right) $$$‐harmonic functions, respectively. For a deeper discussion of these classes of functions, we refer the reader to [7–9, 32, 33].…”

“…A Cauchy-Kowalevski extension theorem for inframonogenic functions appears in [6] and a peculiar connection with the solutions of the Lamé-Navier system in linear elasticity theory is derived in [13,16,17]. More recently in [7][8][9], one defined subclasses of biharmonic functions, which may be understood as somewhat exotic generalization of inframonogenic functions, via the so-called structural sets (to be used mainly in Section 4.2).…”

“…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].…”