A Schwartz-type boundary value problem in a biharmonic plane

S. V. Gryshchuk; S. A. Plaksa

paper

A Schwartz-type boundary value problem in a biharmonic plane

arXiv:1610.00436

Abstract

A commutative algebra $B$ over the field of complex numbers with the bases ${e_{1}, e_{2}}$ satisfying the conditions $(e_{1}^{2} + e_{2}^{2})^{2} = 0$ , $e_{1}^{2} + e_{2}^{2} \neq = 0$ , is considered. The algebra $B$ is associated with the biharmonic equation. Consider a Schwartz-type boundary value problem on finding a monogenic function of the type $Φ (x e_{1} + y e_{2}) = U_{1} (x, y) e_{1} + U_{2} (x, y) i e_{1} + U_{3} (x, y) e_{2} + U_{4} (x, y) i e_{2}$ , $(x, y) \in D$ , when values of two components $U_{1}$ , $U_{4}$ are given on the boundary of a domain $D$ lying in the Cartesian plane $x O y$ . 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.