Hypercomplex Bitsadze system (Q2751145)

Starting with the inhomogeneous complex form of the Cauchy-Riemann system in a plane domain and the representation of its solutions by means of the Cauchy-Pompeiu formula and a suitable analytic function (which may be fixed by some boundary condition) and, besides, with a similar treatment of the Bitsadze equation \(w_{\overline z\overline z}=\widetilde f\), the author sketches how these ideas can be utilized to obtain analogous results in the frame of hypercomplex function theory for elliptic first-order systems which can be transformed to a hypercomplex equation of the form \(Dw=f\) where \(D=\partial_{ \overline z}+ q\partial_z\), NEWLINE\[NEWLINEq=\sum^{n-1}_{k=1} q_ke^k,\;f=\sum^{n-1}_{k=0} f_k e^k,\;w=\sum^{n-1}_{k=0} w_ke^kNEWLINE\]NEWLINE with \(e\) being an \(n\times n\)-matrix satisfying \(e^n=0\). (In the case \(f=0\), \(w\) is called a hyperanalytic function.) As special cases, representations of solutions for the inhomogeneous hypercomplex Bitsadze and Laplace equations are considered. Furthermore, for the inhomogeneous hypercomplex Cauchy-Riemann system, representations of solutions to Schwarz and Riemann-Hilbert problems are obtained. Finally, the solution of a suitable boundary value problem is derived for the hypercomplex Bitsadze system.NEWLINENEWLINEFor the entire collection see [Zbl 0966.00026].