Inframonogenic functions and their applications in 3‐dimensional elasticity theory

Abstract

In this paper, we consider functions satisfying the sandwich equation ∂x_f∂x_=0, where ∂x_ stands for the Dirac operator in Rm. Such functions are referred as inframonogenic and represent an extension of the monogenic functions, ie, null solutions of ∂x_. In particular, for odd m, we prove that a C2‐function is both inframonogenic and harmonic in normalΩ⊂Rm if and only if it can be represented in Ω as f=f1+f2+f3xtrue_+xtrue_f4, where f1 and f2 are, respectively, left and right monogenic functions in Ω, while f3 and f4 are two‐sided monogenic functions there. Finally, in deriving some applications of our results, we have made use of the deep connection between the class of inframonogenic vector fields and the universal solutions of the Lamé‐Navier system in R3.

An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications

“…Recently, several works have addressed the homogeneous Lamé-Navier equation using the tools of quaternionic analysis. For instance, [32,33].…”

Biquaternionic Reformulation of a Fractional Monochromatic Maxwell System

“…where μ > 0 and λ > −2/3μ are the Lamé coefficients (see [36] for more details). The fractional calculus can be used to establish a fractional generalization of nonlocal elasticity in two forms: the fractional gradient elasticity theory (weak nonlocality) and the fractional integral elasticity theory (strong nonlocality) (see [37][38][39]).…”

“…Having in mind the conditions relating λ, μ in (33), it is easily seen that γ ≠ 0 and β ≠ 0. [36] for more details). In this way, the kernel…”

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

“…The solutions of this last equation represent a natural generalization of the so-called inframonogenic functions studied in [7][8][9][10][11], which explains the name of (ϕ, ψ)-inframonogenic functions adopted for them in [2].…”