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

Abstract

A commutative algebra \( \mathbbm{B} \) over the complex field with a basis {e1, e2} satisfying the conditions \( {\left({e}_1^2+{e}_2^2\right)}^2=0,{e}_1^2+{e}_2^2\ne 0 \) 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 Φ (xe1+ye2) = U1(x; y) e1 + U2(x; y) ie1 + U3(x; y) e2 + U4(x; y) ie2, (x; y) ∈ D, when the values of two components—either U1, U3 or U1, U4—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.