Heinrich Begehr

On Riemann boundary value problems for certain linear elliptic systems in the plane

The partial differential equations which occur in the theory of elastic plates and shells are among those which may be reduced to a first order elliptic system of the type studied by Douglis [3]. Under certain regularity conditions for the coefficients, a Beltrami transformation exists takmg the general first-order, elliptic systems into a normal-Douglis-form. This form can be further simplified, and more concisely represented by utilizing the algebra CY of hypercomplex numbers. The theory of solutions to these linear systems is known as generakzed hyperunai’yticfimction theory (see Gilbert [S, 6, 71 and Gilbert and HiIe [g, 9])* and bears the same relationship to the Douglis theory of hyperanalytic functions as Vekua’s theory to analytic functions. In [I] Hilbert boundary value problems for generalized hyperanalytic functions were studied. Subsequently semilinear Douglis systems were treated by Wendland [13] and Gilbert [7j. Th e p resent work deals with Riemann boundary value problems for linear systems. It is clear that our results may be extended to nonlinear hypercomplex systems resembling the complex cases investigated by Warowna-Dorau [14] and Wolska-Bochenek 1151. A good survey of the methods encountered in the analytic case may be found in the monographs of Gakhov [4J and Muskhelishvili [lo]. The fundamental kernels for the linear system (see [S]) permit the formulation of the Riemann boundary value problem for generalized hyperanalytic functions as a Cauchy type integral relation. In general, there is no similarity principle

Boundary value problems for the inhomogeneous polyanalytic equation II

Summary Further mixed boundary value problems are studied for the inhomogeneous polyanalytic equation in the unit disc. As in part I, see [4], the boundary conditions are some combinations of Schwarz, Dirichlet and Neumann conditions. The method applied is based on an iteration process leading from lower order equations to higher ones, see [1–3]. The method can be used to solve all kind of such mixed problems also for other kinds of complex model equations.

Bi-analytic functions of several variables

A function theory of complex Bi-analytic functions in a variables determined by certain second order system is developed. This has been applied to solve the Dirichlet problem of the corresponding systems.

Schwarz problem in lens and lune

The Schwarz problem is explicitly solved for the inhomogeneous Cauchy–Riemann equation in a particular circular lens and two related complementary lunes. The solutions are given by the same analytical formula restricted to the respective domain. The parqueting-reflection method is used to construct the Cauchy–Schwarz representation formula leading to the Schwarz and thus the Poisson kernels for the three domains.

Harmonic Dirichlet problem for some equilateral triangle

The Dirichlet problem for the Poisson equation is explicitly solved in an equilateral triangle of the complex plane.

Randwertaufgaben ganzzahliger charakterjstik für verailgemeinerte hyperanaiytische funktionen1

(1977). Randwertaufgaben ganzzahliger charakterjstik fur verailgemeinerte hyperanaiytische funktionen1. Applicable Analysis: Vol. 6, No. 3, pp. 189-205.

How to find harmonic Green functions in the plane

There are three different methods to determine harmonic Green functions for planar domains. One is based on the Schwarz boundary value problem for analytic functions. Another one is utilizing the conformal invariance of the harmonic Green function. Finally, the Schwarz reflection serves for domains the boundary of which is composed by segments of circles and lines providing a parqueting of the complex plane. Each of these methods is used exemplarily to find the harmonic Green function for particular domains.

Integral Representation Formulas in Polydomains

Representation formulas are given for solutions to second order systems in polydomains composed by as well the Laplace as the Bitsadze operator in the single variables. They result from iterations of the respective formulas for the one variable case. This procedure is also applied to mixed systems of first and second order equations.

Green function for a hyperbolic strip and a class of related plane domains

The parqueting reflection principle serves to construct the harmonic Green function for a hyperbolic strip in the unit disc. The formula obtained represents the Green functions also for the set of strips covering the unit disc and related to the original strip by continued reflection at the boundaries. A complementary unbounded domain is also included. The complex plane is covered by the closures of countably many domains having all the same expression for their Green functions.

Piecewise continuous solutions of pseudoparabolic equations in two space dimensions

Abstract : One of the principal boundary value problems in analytic function theory is the so-called Riemann boundary value problem. The simplest version of the problem requires the finding of an analytic function phi in C/Gamma, where Gamma is a closed smooth contour, and a prescribed Hoelder continuous jump is prescribed for phi across Gamma. The solution of this problem may be given in terms of a Cauchy integral. In generalized analytic, as well as generalized hyperanalytic function theory, a Cauchy-type representation exists, which suggest that the Riemann problem may be solved in a similar way. In the present work several new representations for initial value problems are obtained. An iterative scheme is presented for solving the initial-boundary value problem. These results are of interest for investigating wave motion in anisotropic, nonhomogeneous elastic materials.