Reduction of a Schwartz-type boundary value problem for biharmonic monogenic functions to Fredholm integral equations

Abstract We consider a commutative algebra 𝔹 over the field of complex numbers with a basis {e1, e2} satisfying the conditions (e12+e22)2=0,e12+e22≠0. $ (e_{1}^{2}+e_{2}^{2})^{2}=0, e_{1}^{2}+e_{2}^{2}\neq 0. $ Let D be a bounded simply-connected domain in ℝ2. We consider (1-4)-problem for monogenic 𝔹-valued functions Φ(xe1 + ye2) = U1(x, y)e1 + U2(x, y)i e1 + U3(x, y)e2 + U4(x, y)i e2 having the classic derivative in the domain Dζ = {xe1 + ye2 : (x, y) ∈ D}: to find a monogenic in Dζ function Φ, which is continuously extended to the boundary ∂Dζ, when values of two component-functions U1, U4 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 them
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 fe 1 ; e 2 g satisfying the conditions .e 2 1 C e 2 2 / 2 D 0; e 2 1 C e 2 2 ¤ 0 : 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]): e 1 D 1; e 2 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 . We use the euclidian norm kak WD p jz 1 j 2 C jz 2 j 2 in the algebra B, where a D z 1 e 1 C z 2 e 2 and z 1 ; z 2 2 C. Consider a biharmonic plane fe 1 ; e 2 g WD f D x e 1 C y e 2 W x; y 2 Rg 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 WD f D xe 1 C ye 2 W .x; y/ 2 Dg in the biharmonic plane fe 1 ; e 2 g and the congruent domain D z WD fz D x C iy W .x; y/ 2 Dg in the complex plane C. Its boundaries are denoted by @D, @D and @D z , respectively. Let D (or D z ) be the closure of domain D (or D z ). In what follows, D x e 1 C y e 2 , z D x C iy and x; y 2 R.
We say that a functionˆW D ! B is monogenic in a domain D and denoteˆ2 M B .D /, if the derivativê 0 . / exists at every point 2 D : Everyˆ2 M B .D / has the derivative of any order in D (cf., e.g., [5,6]) and satisfies the equalities due to the conditions (1). Therefore, we shall also term such a functionˆby biharmonic monogenic function in D . Any functionˆW D ! B has an expansion of the typê where U l W D ! R, l D 1; 2; 3; 4, are real-valued component-functions. We shall use the following notation: U l OEˆ WD U l , l D 1; 2; 3; 4.
Everyˆ2 M B .D / is expressed via two corresponding analytic functions F W D z ! C, F 0 W D z ! C of the complex variable z in the form (cf., e.g., [5][6][7]): .
The equality (4) establishes one-to-one correspondence between functionsˆ2 M B .D / and pairs of complexvalued analytic functions F; 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 is also bounded and simply connected.
V. F. Kovalev [8] considered the following boundary value problem for biharmonic monogenic functions: to find a functionˆ2 M B .D / which is continuously extended onto the closure D when values of two componentfunctions in (2) are given on @D , i.e., the following boundary conditions are satisfied: 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][9][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 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][15][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][20][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.

Solving process of (1-4)-problem via analytic functions of the complex variable
A method for solving the (1-4)-problem by means of its reduction to classical Schwartz boundary value problems for analytic functions of the complex variable is delivered in [12,13]. Let us formulate some results of such a kind. For a continuous function u W @D ! R, by b u we denote the function defined on @D z by the equality The following theorem is proved in the papers [12,13].
Theorem 2.1. Let u l W @D ! R, l 2 f1; 4g, are continuous functions and, moreover, a function yF 0 .z/ has a continuous extendibility to the boundary @D z , where F is a solution of the classical Schwartz problem with the boundary condition Then a solution of the (1-4)-problem is expressed by the formula (4), where a function F 0 is a solution of the classical Schwartz problem with the boundary condition A particular case of Theorem 2.1 is the following theorem. Note, that Theorem 2.2 is proved in [12,13] for an arbitrary (bounded or unbounded) simply connected domain D .

Scheme for reducing the (1-4)-problem to a system of Fredholm integral equations
Let us develop a method for solving the inhomogeneous (1-4)-problem without an essential in Theorem 2.1 assumption that a function yF 0 .z/ has a continuous extendibility to the boundary @D z . Let the boundary @D be a closed smooth Jordan curve. We assume that every boundary function u l , l 2 f1; 4g, satisfies the Dini condition with the modulus of continuity !.u l ; "/ WD sup We seek solutions in a class of functions represented in the form where the functions ' l W @D ! R, l 2 f1; 4g, satisfy Dini conditions of the type (5). It is proved in Theorem 4.2 [10] that the integral (6) .
where '. / WD ' 1 . /e 1 C ' 4 . /i e 2 and a singular integral is understood in the sense of its Cauchy principal value: We use a conformal mapping z D .t / of the upper half-plane ft 2 C W Im t > 0g onto the domain D z . Denote Inasmuch as the mentioned conformal mapping is continued to a homeomorphism between the closures of corresponding domains, the function e .s/ WD 1 .s/e 1 C 2 .s/e 2 8 s 2 R generates a homeomorphic mapping of the extended real axis R WD R [ f1g onto the curve @D . Introducing the function g.s/ WD g 1 .s/e 1 C g 4 .s/i e 2 8 s 2 R ; where g l .s/ WD ' l .e .s//, l 2 f1; 4g, we rewrite the equality (7) in the form (cf. [10]) and a correspondence between the points 0 2 @D n fe .1/g and t 2 R is given by the equality 0 D e .t /. Now, we single out components U l OEˆC. 0 /, l 2 f1; 4g, and after the substitution them into the boundary conditions of the (1-4)-problem, we shall obtain the following system of integral equations for finding the functions g 1 and g 4 : (9) where e u l .t / WD u l e .t / , l 2 f1; 4g.
Let C.R / denote the Banach space of functions g W R ! C that are continuous on the extended real axis R with the norm kg k C.R / WD sup t 2R jg .t /j. In Theorem 6.13 [10] there are conditions which are sufficient for compactness of integral operators in equations of the system (9) in the space C.R /.
To formulate such conditions, consider the conformal mapping .T / of the unit disk fT 2 C W jT j < 1g onto for all t 2 ft 2 C W Im t > 0g. Thus, it follows from Theorem 6.13 [10] that if the conformal mapping .T / have the nonvanishing continuous contour derivative 0 .T / on the unit circle WD fT 2 C W jT j D 1g, and its modulus of continuity ! . 0 ; "/ WD sup T 1 ;T 2 2; jT 1 T 2 jÄ" satisfies a condition of the type (5), then the integral operators in the system (9) are compact in the space C. R /. Let D.R/ denote the class of functions g 2 C.R / whose moduli of continuity ! R .g ; "/ D sup 1 ; 2 2RWj 1 2 jÄ" jg . 1 / g . 2 /j ; ! R;1 .g ; "/ D sup 2RWj j 1=" jg . / g .1/j satisfy the Dini conditions Since the functions ' 1 , ' 4 in the expression (6) of a solution of the (1-4)-problem have to satisfy conditions of the type (5), it is necessary to require that corresponding functions g 1 , g 4 satisfying the system (9) should belong to the class D.R/. In the next theorem we state a condition on the conformal mapping .T /, under which all solutions of the system (9) satisfy the mentioned requirement. Then the following assertions are true: (i) the system of Fredholm integral equations (9) has the unique solution in C. R /; (ii) all functions g 1 ; g 4 2 C. R / satisfying the system (9) belong to the class D.R/, and the corresponding functions ' 1 , ' 4 in (6) satisfy Dini conditions of the type (5).
Proof. In such a way as in Theorem 7.2 [10], we establish that all functions g 1 ; g 4 2 C. R / satisfying the system (9) belong to the class D.R/. Thus, corresponding functions ' 1 , ' 4 in (6) satisfy Dini conditions of the type (5).
To prove that the system (9) has the unique solution, consider the homogeneous system of equations (9)  Then the function, which is defined for all 2 D by the formulâ is a solution of the homogeneous (1-4)-problem with u 1 D u 4 Á 0 that is corresponding to the homogeneous system of equations (9). Taking into account Theorem 2.2, we have the equalityˆ0. / D a 1 i e 1 C a 2 e 2 for all 2 D . Therefore, from the Sokhotski-Plemelj formulas (7), (8) we obtain the following relations: which imply the equalitŷ Thus, the function (10) is monogenic in fe 1 ; e 2 g n D and satisfies the equalities Now, taking into account Theorem 2.2, which is undoubtedly true for unbounded domain fe 1 ; e 2 g n D , one can easily conclude thatˆ0 is constant in fe 1 ; e 2 g n D . Moreover,ˆ0. / Á 0 in fe 1 ; e 2 g n D because the function (10) is vanishing at infinity. At last, after a substitution of 0 into the left-hand side of equality (11) we conclude that g 0 1 D g 0 4 Á 0 , i.e. the homogeneous system of equations (9) has only the trivial solution. Therefore, by the Fredholm theory, the non-homogeneous system of equations (9) has a unique solution.

A displacements-type boundary value problem
A relation between the (1-4)-problem and a displacements-type boundary value problem of the plane elasticity theory is considered in [17,18].
Consider the following displacements-type boundary value .u x ; v y /-problem: to find in a domain D of the Cartesian plane xOy the first order partial derivatives V 1 WD @u @x , V 2 WD @v @y for displacements u D u.x; y/, v D v.x; y/ of an elastic isotropic body occupying D, when their limiting values are given on @D: lim .x;y/!.x 0 ;y 0 /; .x;y/2D V k .x; y/ D h k .x 0 ; y 0 / 8 .x 0 ; y 0 / 2 @D; k D 1; 2; where h k W @D ! R, k D 1; 2, are given functions.
Taking into account the equalities (12), one can conclude that the .u x ; v y /-problem is equivalent to finding the functions C k .x; y/, k D 1; 2, in the domain D, when their boundary values satisfy the conditions C k .x 0 ; y 0 / D 2 h k .x 0 ; y 0 / 8.x 0 ; y 0 / 2 @D; k D 1; 2: (13) Assuming that W has continuous partial derivatives till the second order up to the boundary @D, we deduce that in this case .u x ; v y /-problem is reduced to finding the functions W k , k D 1; 2, in the domain D, when their limiting values are given on @D.
Consider in the domain D a functionˆ 2 M B .D / for which U 1 OEˆ D W . Then it follows from the Cauchy-Riemann condition (3) that for the biharmonic monogenic functionˆWDˆ0 0 the following equalities hold: U 1 OEˆ. / D W 1 .x; y/; U 4 OEˆ. / D 1 2 W 1 .x; y/ W 2 .x; y/ Á : Thus, in such a way .u x ; v y /-problem is reduced to (1-4)-problem with appropriate boundary conditions, and Theorem 3.1 makes it possible to find the elastic equilibrium for an isotropic body occupying D. In this case, for finding stresses an additional assumption on the function W is required, but finding displacements is free of any assumptions (see [18]).