An 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 {*e*_{1}, *e*_{2}} satisfying the conditions
$$({e}_{1}^{2}+{e}_{2}^{2}{)}^{2}=0,\phantom{\rule{2em}{0ex}}{e}_{1}^{2}+{e}_{2}^{2}\ne 0.$$(1)

Such a basis {*e*_{1}, *e*_{2}} is also called *biharmonic*.

In the paper [2] I. P. Melľnichenko 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 {*e*_{1}, *e*_{2}} with the following multiplication table (see [1]):
$${e}_{1}=1,\phantom{\rule{2em}{0ex}}{e}_{2}^{2}={e}_{1}+2i{e}_{2},$$

where *i* is the imaginary complex unit. We consider also a basis {1, *ρ*} (see [2]), where a nilpotent element *ρ*= 2*e*_{1} + 2*i e*_{2} satisfies the equality *ρ*^{2} = 0.

We use the euclidian norm
$\parallel a\parallel :=\sqrt{|{z}_{1}{|}^{2}+|{z}_{2}{|}^{2}}$ in the algebra 𝔹, where *a* = *z*_{1}*e*_{1} + *z*_{2}*e*_{2} and *z*_{1}, *z*_{2} ∈ ℂ. Consider a *biharmonic plane* *μ*{*e*_{1}, *e*_{2}} := {*ζ* = *xe*_{1} + *ye*_{2} : *x*, *y* ∈ ℝ} which is a linear span of the elements *e*_{1}, *e*_{2} over the field of real numbers ℝ. With a domain *D* of the Cartesian plane *xOy* we associate the congruent domain *D*_{ζ} := {*ζ* = *xe*_{1} + *ye*_{2} : (*x*, *y*) ∈ *D*} in the biharmonic plane *μ*{*e*_{1}, *e*_{2}} and the congruent domain *D*_{Z} := {*z* = *x* + *iy* : (*x*, *y*) ∈ *D*} in the complex plane ℂ. Its boundaries are denoted by ∂*D*, ∂*D*_{ζ} and ∂*D*_{Z}, respectively. Let
$\overline{{D}_{\zeta}}\phantom{\rule{thinmathspace}{0ex}}(\text{or}\phantom{\rule{thinmathspace}{0ex}}\overline{{D}_{Z}})$ be the closure of domain *D*_{ζ} (or *D*_{Z}). In what follows, *ζ* = *xe*_{1} + *ye*_{2}, *z* = *x* + *iy* and *x*, *y* ∈ ℝ.

We say that a function Φ : *D*_{ζ} → 𝔹 is *monogenic* in a domain *D*_{ζ} and denote Φ ∈ 𝓜_{𝔹}(*D*_{ζ}), if the derivative Φ′(*ζ*) exists at every point *ζ* ∈ *D*_{ζ}:
$${\mathrm{\Phi}}^{\prime}(\zeta ):=\underset{h\to 0,\phantom{\rule{thinmathspace}{0ex}}h\in \mu \{{e}_{1},{e}_{2}\}}{lim}(\mathrm{\Phi}(\zeta +h)-\mathrm{\Phi}(\zeta )){h}^{-1}.$$

Every Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) has the derivative of any order in *D*_{ζ} (cf., e.g., [5, 6]) and satisfies the equalities
$$({\mathrm{\Delta}}_{2}{)}^{2}\mathrm{\Phi}(\zeta )\equiv (\frac{{\mathrm{\partial}}^{4}}{\mathrm{\partial}{x}^{4}}+2\frac{{\mathrm{\partial}}^{4}}{\mathrm{\partial}{x}^{2}\mathrm{\partial}{y}^{2}}+\frac{{\mathrm{\partial}}^{4}}{\mathrm{\partial}{y}^{4}})\mathrm{\Phi}(\zeta )={\mathrm{\Phi}}^{(4)}(\zeta )({e}_{1}^{2}+{e}_{2}^{2}{)}^{2}=0\phantom{\rule{1em}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\zeta \in {D}_{\zeta}.$$

due to the conditions (1).

Therefore, we shall also term such a function Φ by *biharmonic monogenic* function in *D*_{ζ}. Any function Φ: *D*_{ζ} → 𝔹 has an expansion of the type
$$\mathrm{\Phi}(\zeta )={U}_{1}(x,y){e}_{1}+{U}_{2}(x,y)i{e}_{1}+{U}_{3}(x,y){e}_{2}+{U}_{4}(x,y)i{e}_{2},$$(2)

where *U*_{l}: *D* → ℝ, *l* = 1, 2, 3, 4, are real-valued component-functions. We shall use the following notation: *U*_{l}[Φ] := *U*_{l}, *l* = 1, 2, 3, 4.

All component-functions *U*_{l}, *l* = 1, 2, 3, 4, in the expansion (2) of any Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) are biharmonic functions, i.e., satisfy the biharmonic equation (Δ_{2})^{2}*U*(*x*, *y*)= 0 in *D*. At the same time, every biharmonic in a simply-connected domain *D* function *U*(*x*, *y*) is the first component *U*_{1} ≡ *U* in the expression (2) of a certain function Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) and, moreover, all such functions Φ are found in [5, 6] in an explicit form.

It is proved in [1] that Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) if and only if its each real-valued component-function in (2) is real differentiable in *D* and the following analog of the Cauchy – Riemann condition is fulfilled:
$$\frac{\mathrm{\partial}\mathrm{\Phi}(\zeta )}{\mathrm{\partial}y}{e}_{1}=\frac{\mathrm{\partial}\mathrm{\Phi}(\zeta )}{\mathrm{\partial}x}{e}_{2}\phantom{\rule{1em}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\zeta \in {D}_{\zeta}.$$(3)

Every Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) is expressed via two corresponding analytic functions *F* : *D*_{Z} → ℂ, *F*_{0}: *D*_{Z} → ℂ of the complex variable *z* in the form (cf., e.g., [5–7]):
$$\mathrm{\Phi}(\zeta )=F(z){e}_{1}-(\frac{iy}{2}{F}^{\prime}(z)-{F}_{0}(z))\rho \phantom{\rule{1em}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\zeta \in {D}_{\zeta}.$$(4)

The equality (4) establishes one-to-one correspondence between functions Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) and pairs of complex-valued analytic functions *F*, *F*_{0} in *D*_{Z}.

In what follows, we assume that the domain *D*_{Z} is a bounded and simply-connected, and in this case we shall say that the domain *D*_{ζ} is also bounded and simply connected.

V. F. Kovalev [8] considered the following boundary value problem for biharmonic monogenic functions: to find a function Φ ∈ 𝓜_{𝔹}(*D*_{ζ}) which is continuously extended onto the closure
$\overline{{D}_{\zeta}}$ when values of two component-functions in (2) are given on ∂*D*_{ζ}, i.e., the following boundary conditions are satisfied:
$${U}_{k}(x,y)={u}_{k}(\zeta ),\phantom{\rule{1em}{0ex}}{U}_{m}(x,y)={u}_{m}(\zeta )\phantom{\rule{1em}{0ex}}\mathrm{\forall}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\zeta \in \mathrm{\partial}{D}_{\zeta},\phantom{\rule{1em}{0ex}}1\le k<m\le 4,$$

where *u*_{k} and *u*_{m} are given functions.

We shall call such a problem by the (*k*-*m*)-problem. V. F. Kovalev [8] called it by a *biharmonic Schwartz problem* owing to its analogy with the classical Schwartz problem on finding an analytic function of the complex variable when values of its real part are given on the boundary of domain.

It was established in [8] that all biharmonic Schwartz problems are reduced to the main three problems: the (1-2)-problem or the (1-3)-problem or the (1-4)-problem.

It is shown (see [8–10]) that the fundamental biharmonic problem (cf, e.g., [11, p.146]) is reduced to the (1-3)- problem. In [9], we investigated the (1-3)-problem for cases where *D*_{ζ} is either an upper half-plane or a unit disk in the biharmonic plane. Its solutions were found in explicit forms with using of some integrals analogous to the classic Schwarz integral. Similar results are obtained in [12, 13] for the (1-4)-problem.

In [10], using a hypercomplex analog of the Cauchy type integral, we reduced the (1-3)-problem to a system of integral equations and established sufficient conditions under which this system has the Fredholm property. It was made for the case where the boundary of domain belongs to a class being wider than the class of Lyapunov curves that was usually required in the plane elasticity theory (cf., e.g., [11, 14–16]).

In this paper we develop a method for reducing (1-4)-problem to a system of the Fredholm integral equations. The obtained results are appreciably similar to respective results for (1-3)-problem in [10], however, in contrast to (1-3)-problem, which is solvable in a general case if and only if a certain natural condition is satisfied, the (1-4)-problem is solvable unconditionally.

Let us underscore that (1-4)-problem is used in [17, 18] for solving a displacements-type boundary value problem on finding some partial derivatives of displacements by their limiting values in the case of 2-D isotropic elasticity theory.

Note that in the papers [4, 19–21] for investigations of biharmonic functions there are other approaches, which involve commutative finite-dimensional algebras over the field ℝ and suitable “analytic” hypercomplex functions.

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