A Schwartz-type boundary value problem in a biharmonic plane

Abstract

A commutative algebra B over the field of complex numbers with the bases {e₁, e₂} satisfying the conditions (e₁² + e₂²)² = 0, e₁² + e₂²)² ≠ 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(xe₁ + ye₂) = U₁(x, y)e₁ + U₂(x, y)ie₁ + U₃(x, y)e₂ + U₄(x, y)ie₂, (x, y) ∈ D, when values of two components U₁, U₄ 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.