A Schwartz-type boundary value problem in a biharmonic plane

Abstract

A commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$, is considered. The algebra $\mathbb{B}$ is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem 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 values of two components $U_1$, $U_4$ are given on the boundary of a domain $D$ lying in the Cartesian plane $xOy$. We develop a method of its solving which is based on expressions of monogenic functions via corresponding analytic functions of the complex variable. For a half-plane and for a disk, solutions are obtained in explicit forms by means of Schwartz-type integrals.