Monogenic functions in the biharmonic boundary value problem

We consider a commutative algebra over the field of complex numbers with a basis {e1,e2} satisfying the conditions , . Let D be a bounded domain in the Cartesian plane xOy and Dζ={xe1+ye2:(x,y)∈D}. Components of every monogenic function Φ(xe1+ye2) = U1(x,y)e1+U2(x,y)ie1+U3(x,y)e2+U4(x,y)ie2 having the classic derivative in Dζ are biharmonic functions in D, that is, Δ2Uj(x,y) = 0 for j = 1,2,3,4. We consider a Schwarz-type boundary value problem for monogenic functions in a simply connected domain Dζ. This problem is associated with the following biharmonic problem: to find a biharmonic function V(x,y) in the domain D when boundary values of its partial derivatives ∂V/∂x, ∂V/∂y are given on the boundary ∂D. Using a hypercomplex analog of the Cauchy-type integral, we reduce the mentioned Schwarz-type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright