On initial value problems for quaternionic valued functions

Abstract

In [2], [6], [7], methods are discussed for solving initial value problems

$$\left\{ \begin{gathered} \frac{{\partial u}}{{\partial t}}\left( {t,x_0 ,x_1 } \right) = f\left( {t,x_0 ,x_1 ,u\left( {t,x_0 ,x_1 } \right),\frac{{\partial u}}{{\partial x_0 }},\frac{{\partial u}}{{\partial x_1 }}} \right) \hfill \\ u\left( {0,x_0 ,x_1 } \right) = u^{\left( 0 \right)} \left( {x_0 ,x_1 } \right) \hfill \\ \end{gathered} \right.$$

in certain scales of Banach spaces. The crucial point is to use suitable interior estimates for the complex valued functionu=u ₀+iu ₁. For holomorphic functionsu these estimates follow from Cauchy’s integral formula or from equivalent estimates for the harmonic partsu ₀ andu ₁.

In this paper we consider the (linear) case for quaternionic-valued functionsu=u ₀ e ₀+u ₁ e ₁+u ₁ e ₂+u ₃ e ₃,u _i=u _i (t,x ₀,x ₁,x ₂,x ₃), by transferring the real-valued 4×4 system to an equivalent quaternionic equation and dealing with monogenic solutions. Finally we consider a special Dirac system for a certain non-monogenic case.