Band 30, Heft 2

The Lame system of general anisotropic plane elasticity is considered. A representation of a general solution or the system through a so-called Douglis analytic functions is given. The cases of orthotropic and isotropic media are also considered.

In [1–10], the authors posed and discussed the Tricomi problem of second order equations of mixed (elliptic-hyperbolic) type, which possesses important applications to gas dynamics, but they only consider some special equations of mixed type. In [5, 7], the authors proposed and discussed the Tricomi problem for some second order equations of mixed type with nonsmooth degenerate line under stronger conditions. The present paper deals with the oblique derivative problem for second order nonlinear equations of mixed type with two parabolic degenerate lines in a plane domain. The boundary value problem includes the Tricomi problem of Chaplygin equation as a special case. Firstly the existence of solutions of corresponding boundary value problems for degenerate elliptic and hyperbolic equations of second order are discussed, and then the solvability of the oblique derivative problem for the equations of mixed type with two degenerate lines is proved. The used method in this paper is different to those in above papers, because the equations are nonlinear, and the new notations are introduced, such that the second order equations of mixed type with degenerate lines are reduced to the first order mixed complex equations with singular coefficients, then the advantage of complex analytic method can be applied, and we can see that the method can be used to handle more general problems. Finally we mention that the authors in [21–24] discussed oblique derivative problems for the Helmholtz and Laplace equations in the general interior and exterior domains.

In this paper, we obtain some results in the spirit of Gamma-lines theory (complex analysis) for large classes of real functions f ( x,y ). In Section 3 our main result describe the length of solutions of f ( x,y )= A ν , for arbitrary finite sets of values A ν . This leads to some consequences for real algebraic polynomials P ( x,y ) in Section 1 and modules | p ( z )| of complex polynomials p ( z ) in Section 2.

Explicit expressions for the Green´s function and Poisson kernel for quarter space and octants in the Heisenberg group H 1 are obtained for circular boundary data. Using Poisson kernel, inhomogeneous Dirichlet problem is discussed on H 1 . Explicit Green´s function for the powers of Laplacian in case of Korányi ball is also given.

In this paper, we introduce the operators related to mixed boundary value problems for general linear elliptic partial complex differential equations in the unit disc of the complex plane. The solvability of the relevant boundary value problems will be studied by transforming them into singular integral equations.

The aim of this paper is the construction of a measure of the rate of convergence or the rate that a function converges to a limit value, in the category of real functions. The potential convergence, as it will be defined later on, constitutes the measure of the rate of convergence, while its calculation is done through a suitable algorithm. This theory applies in the field of ordinary differential equations suggesting methods of tracking down and finding particular solutions on the class of non linear D.E.

We study the connection between multiplicities of the zeros and boundary behavior of bounded analytic and harmonic functions. We prove existence of angular (non-tangential) limit at a boundary point provided that multiplicities of zeroes of the function grow fast enough on a given sequence of points approaching the boundary.