Differential Operators Associated to the Cauchy-Fueter Operator in Quaternion Algebra

Abstract

This paper deals with the initial value problem of the type

$${\frac{\partial{w}}{\partial{t}}}=L\left(t,x,w,{\frac{\partial{w}}{\partial{x_i}}}\right)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(1)$$

$$w(0,x)=\varphi(x)\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(2)$$

where t is the time, L is a linear first order operator (matrix-type) in Quaternionic Analysis and ${\varphi}$ is a regular function taking values in the Quaternionic Algebra. The article proves necessary and sufficient conditions on the coefficients of operator L under which L is associated to the Cauchy-Fueter operator of Quarternionic Analysis.

This criterion makes it possible to construct the operator L for which the initial problem (1), (2) is solvable for an arbitrary initial regular function ${\varphi}$ and the solution is also regular for each t.