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 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 .

Schwartz-Type Boundary Value Problems for Monogenic Functions in a Biharmonic Algebra

“…Below, we show a method for reducing (1-3)-problem and (1-4)-problem to systems of the Fredholm integral equations. Such a method was developed in [9,12]. Obtained results are appreciably similar for the mentioned problems, 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.…”

“…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., [14][15][16][17][18]). The similar is done for the (1-4)-problem in [12].…”

“…He named such problems as biharmonic Schwartz problems owing to their analogy with the classic Schwartz problem on finding an analytic function of a complex variable when values of its real part are given on the boundary of domain. We shall call problems of such a type as (k-m)-problems in the sense of Kovalev. Note, that in previous papers [5,6,[8][9][10][11][12][13] we interpret the (k-m)-problem as the (k-m)-problem in the sense of Kovalev. It was established in [7] that all (k-m)-problems are reduced to the main three problems: with k = 1 and m ∈ {2, 3, 4}, respectively.…”

“…In [12,13], there is considered a relation between (1-4)-problem and boundary value problems of the plane elasticity theory. Namely, there is considered a problem on finding an elastic equilibrium for isotropic body occupying D with given limiting values of partial derivatives ∂u ∂x , ∂v ∂y for displacements u = u(x, y) , v = v(x, y) on the boundary ∂D.…”

Differentiable functions in a three-dimensional associative noncommutative algebra

“…Integral representations of the dierentiable in some sense, functions are important for applications, especially for solving boundary value problems. For example, in the papers [13,14] integral representation of monogenic functions in the two-dimensional biharmonic algebra were applied to solving boundary value problems for two-dimensional biharmonic equation. In the monograph [15] integral representations of axialsymmetric potential and Stokes ow functions were obtained and such representations were applied to the solution of Dirichlet boundary problem for the mentioned potentials.…”