Schwartz-type boundary-value problems for canonical domains in a biharmonic plane

Abstract

A commutative algebra $\mathbb{B}$ over the complex field with a basis $% \{e_{1},e_{2}\}$ satisfying the conditions $(e_{1}^{2}+e_{2}^{2})^{2}=0$, $% e_{1}^{2}+e_{2}^{2}\neq 0$ is considered. This algebra is associated with the 2-D biharmonic equation. We consider Schwartz-type boundary-value problems on finding a monogenic function of the type $\Phi (xe_{1}+ye_{2})=U_{1}(x,y)\,e_{1}+U_{2}(x,y)\,ie_{1}+U_{3}(x,y)% \,e_{2}+U_{4}(x,y)\,ie_{2}$, $(x,y)\in D$, when the values of two components---either $U_{1}$, $U_{3}$ or $U_{1}$, $U_{4}$---are given on the boundary of a domain $D$ lying in the Cartesian plane $xOy$. For solving those boundary-value problems for a half-plane and for a disk, we develop methods that are based on solution expressions via Schwartz-type integrals and obtain solutions in the explicit form.