$\mathbb{B}$-valued monogenic functions and their applications to boundary value problems in displacements of 2-D Elasticity

Abstract

Consider the commutative algebra $\mathbb{B}$ over the field of complex numbers with the bases $\{e_1,e_2\}$ such that %satisfying the conditions $(e_1^2+e_2^2)^2=0$, $e_1^2+e_2^2\ne 0$. %$\mathbb{B}$ is unique. Let $D$ be a domain in $xOy$, $D_{\zeta}:=\{xe_1+ye_2:(x,y) \in D\}\subset \mathbb{B}$. We say that $\mathbb{B}$-valued function $\Phi \colon D_{\zeta} \longrightarrow \mathbb{B}$, $\Phi(\zeta)=U_{1}\,e_1+U_{2}\,ie_1+ U_{3}\,e_2+U_{4}\,ie_2$, $\zeta=xe_1+ye_2$, $U_{k}=U_{k}(x,y)\colon D\longrightarrow \mathbb{R}$, $k=\bar{1,4}$, is {\em monogenic} in $D_{\zeta}$ iff $\Phi$ has the classic derivative in every point in $D_{\zeta}$. Every $U_k$, $k=\bar{1,4}$, is a biharmonic function in $D$. A problem on finding an elastic equilibrium for isotropic body $D$ by given boundary values on $\partial D$ of partial derivatives $\frac{\partial u}{\partial v}$, $\frac{\partial v}{\partial y}$ for displacements $u$, $v$ is equivalent to BVP for monogenic functions, which is to find $\Phi$ by given boundary values of $U_1$ and $U_4$.