Monogenic functions in the biharmonic boundary value problem

Abstract

We consider a commutative algebra $\mathbb{B}$ over the field of complex numbers with a basis $\{e_1,e_2\}$ satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. Let $D$ be a bounded domain in the Cartesian plane $xOy$ and $D_{\zeta}=\{xe_1+ye_2 : (x,y)\in D\}$. Components of every monogenic function $\Phi(xe_1+ye_2)=U_{1}(x,y)\,e_1+U_{2}(x,y)\,ie_1+ U_{3}(x,y)\,e_2+U_{4}(x,y)\,ie_2$ having the classic derivative in $D_{\zeta}$ are biharmonic functions in $D$, i.e. $\Delta^{2}U_{j}(x,y)=0$ for $j=1,2,3,4$. We consider a Schwarz-type boundary value problem for monogenic functions in a simply connected domain $D_{\zeta}$. 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 $\partial V/\partial x$, $\partial V/\partial y$ are given on the boundary $\partial D$. 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.