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.Comment: 30 page

We consider the commutative algebra B over the field of complex numbers with the bases {e1, e2} satisfying the conditions (e 2 1 + e 2 2 ) 2 = 0 , e 2 1 + e 2 2 = 0 . The algebra B is associated with the biharmonic equation. For monogenic functions with values in B , we consider a Schwartz-type boundary value problem (associated with the main biharmonic problem) for a half-plane and for a disk of the biharmonic plane {xe1 + ye2} , where x, y are real. We obtain solutions in explicit forms by means of Schwartz-type integrals and prove that the mentioned problem is solvable unconditionally for a half-plane but it is solvable for a disk if and only if a certain natural condition is satisfied.

We consider monogenic functions given in a biharmonic plane and taking values in a commutative algebra associated with the biharmonic equation. For the mentioned functions, we establish basic properties analogous to properties of holomorphic functions of the complex variable: the Cauchy integral theorem and integral formula, the Morera theorem, the uniqueness theorem, the Taylor and Laurent expansions.

: We consider a commutative algebra B over the field of complex numbers with a basis fe 1 ; e 2 g satisfying the conditions .e (1-4)-problem for monogenic B-valued functionsˆ.xe 1 C ye 2 / D U 1 .x; y/ e 1 C U 2 .x; y/ i e 1 C U 3 .x; y/ e 2 C U 4 .x; y/ i e 2 having the classic derivative in the domain D D fxe 1 C ye 2 W .x; y/ 2 Dg: to find a monogenic in D functionˆ, which is continuously extended to the boundary @D , when values of two component-functions U 1 , U 4 are given on the boundary @D. Using a hypercomplex analog of the Cauchy type integral, we reduce the (1-4)-problem to a system of integral equations on the real axes. We establish sufficient conditions under which this system has the Fredholm property and the unique solution. We prove that a displacements-type boundary value problem of 2-D isotropic elasticity theory is reduced to (1-4)-problem with appropriate boundary conditions. Biharmonic monogenic functions and Schwarz-type boundary value problems for themAn associative commutative two-dimensional algebra B with the unit 1 over the field of complex numbers C is called biharmonic (see [1,2]) if in B there exists a basis fe 1 ; e 2 g satisfying the conditions.e Such a basis fe 1 ; e 2 g is also called biharmonic. In the paper [2] I. P. Mel'nichenko proved that there exists the unique biharmonic algebra B, and he constructed all biharmonic bases in B. Note that the algebra B is isomorphic to four-dimensional over the field of real numbers R algebras considered by A. Douglis [3] and L. Sobrero [4].In what follows, we consider a biharmonic basis fe 1 ; e 2 g with the following multiplication table (see [1]):2 D e 1 C 2i e 2 ; where i is the imaginary complex unit. We consider also a basis f1; g (see [2]), where a nilpotent element D 2e 1 C 2i e 2 satisfies the equality 2 D 0 .