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 $D_{\zeta }=\{x{e}_1+y{e}_2:(x,y)\in D\}$. Components of every monogenic function $\Phi (x{e}_1+y{e}_2)=U_1(x,y)e_1+U_2(x,y)i{e}_1+U_3(x,y)e_2+U_4(x,y)i{e}_2$ 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.

30 pages

References in corpus (1)

• Schwartz-type integrals in a biharmonic plane

Cited by in corpus (2)

• $\mathbb{B}$-valued monogenic functions and their applications to boundary value problems in displacements of 2-D Elasticity
• ${\mathbb{B}}_0$-Valued Monogenic Functions to the theory of Plane Anisotropy