Complex Equations Of First Order Research Articles

The Riemann-Hilbert problem for mixed complex equations of first order with degenerate rank 0

This article deals with the Riemann-Hilbert boundary value problem for quasilinear mixed (ellipti-chyperbolic) complex equations of first order with degenerate rank 0. Firstly, we give the representation theorem and prove the uniqueness of solutions for the boundary value problem. Afterwards, by using the method of successive iteration, the existence and estimates of solutions for the boundary value problem are verified. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order. In this article, the proof of Holder continuity of a singular double integer is very difficult and interesting as in this Section 4 below.

THE NEW WELL-POSEDNESS OF DISCONTINUOUS RIEMANN-HILBERT PROBLEM FOR ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER WITH TWO DEGENERATE LINES

This article deals with some elliptic complex equations of first order, i.e. the generalized Beltrimi equation with two degenerate lines in the discussed multiply connected domain. We first propose the new well-posed-ness of discontinuous Riemman-Hilbert problem, give estimates of solutions for the modified boundary value problem. Afterwards by using the method of parame- ter extension, the existence of continuous solutions for the generalized Beltrimi equation is verified. In the article, the proof of Hlder continuity of a singular double integer is very difficult and interesting. The above problem possesses the important applications to the Tricomi problem of mixed type equations of second order.

A New Well-Posed Version of Riemann–Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Domains

In this article, we first propose the Riemann–Hilbert problem for uniformly elliptic complex equations of first order in multiply connected domains and its new well-posed-ness with continuous solutions. Then we obtain a priori estimates of solutions of the modified Riemann–Hilbert problem, and derive the solvability results of the modified and original Riemann–Hilbert problems. Finally we shall verify the equivalence of two kinds of well-posed-ness for the Riemann–Hilbert boundary value problem.

The discontinuous Riemann-Hilbert problem for elliptic complex equations of first order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann-Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.

The Riemann-Hilbert problem for degenerate elliptic complex equations and its application

This article deals with some elliptic complex equations of first order, i.e. the generalized Beltrami equation with two degenerate lines in multiply connected domains. We first propose the Riemann–Hilbert problem and give estimates of solutions for the boundary value problem. Afterwards by using the method of parameter extension, the existence of continuous solutions for the generalized Beltrami equation is verified. Finally, we shall prove the existence of a homeomorphic solution of the Beltrami equation with two degenerate lines from the multiply connected domain D onto the rectilinear slit domain. In the article, the Hlder continuity of a singular double integral is applied. The above problem possesses important applications to boundary value problem of mixed type equations.

Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains

In this article, we discuss the approximate method of solving the Riemann-Hilbert boundary value problem for nonlinear uniformly elliptic complex equation of first order (0.1) with the boundary conditions (0.2) in a multiply connected unbounded domain D, the above boundary value problem will be called Problem A. If the complex Equation (0.1) satisfies the conditions similar to Condition C of (1.1), and the boundary condition (0.2) satisfies the conditions similar to (1.5), then we can obtain approximate solutions of the boundary value problems (0.1) and (0.2). Moreover the error estimates of approximate solutions for the boundary value problem is also given. The boundary value problem possesses many applications in mechanics and physics etc., for instance from (5.114) and (5.115), Chapter VI, [1], we see that Problem A of (0.1) possesses the important application to the shell and elasticity.

Boundary value problems for first-order mixed complex equations with degenerate curve

This article deals with some boundary value problems for quasilinear mixed (elliptic-hyperbolic) complex equations of first order with degenerate hyperbolic curve in a simply connected domain. First, we give the representation theorem and prove the uniqueness of solutions for the above boundary value problems. Then using the method of successive iteration, the existence of solutions for the above problems is proved.

Riemann-Hilbert problem for first order complex equations of mixed type with degenerate curve

This paper considers the Riemann-Hilbert problem for linear mixed (elliptic-hyperbolic) complex equations of first order with degenerate curve in a simply connected domain. We first give the representation theorem and uniqueness of solutions for such boundary value problem. Then by using the methods of successive iteration and parameter extension, the existence of solutions for this problem is proved.

Some boundary value problems for nonlinear degenerate elliptic equations of second order

The present article deals with some boundary value problems for nonlinear elliptic equations with degenerate rank 0 including the oblique derivative problem. Firstly the formulation and estimates of solutions of the oblique derivative problem are given, and then by the above estimates and the method of parameter extension, the existence of solutions of the above problem is proved. In this article, the complex analytic method is used, namely the corresponding problem for degenerate elliptic complex equations of first order is firstly discussed, afterwards the above problem for the degenerate elliptic equations of second order is solved.

The mixed boundary‐value problem for second order elliptic equations with degenerate curve on the sides of an angle

AbstractIn [1, 2, 3, 4, 6], the authors posed and discussed some boundary‐value problems of second order elliptic equations with parabolic degeneracy. In [5], the authors posed and discussed some boundary‐value problems of second order mixed equations with degenerate curve on the sides of an angle, but the coefficients of the equations possess strong restrictions. The present paper deals with the mixed problem for elliptic equations with degenerate curve on the sides of an angle, where the coefficients satisfy general conditions. We first give the formulation of the problem and estimates of solutions of the problem for the equations, and then prove the existence of solutions for the above problem by the Leray–Schauder theorem. In this paper, we use the complex analytic method, namely we first introduce the new notation (2.1) below and reduce the degenerate elliptic equations of second order to the corresponding problem for degenerate elliptic complex equations of first order, afterwards the above problem of second order degenerate elliptic equations can be solved. The results in this paper will be used in a subsequent paper to handle the Tricomi problem of second order equations of mixed type with degenerate curve on the sides of an angle. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Estimates of solutions for degenerate elliptic equations of second order§

This article deals with some boundary value problems for elliptic equations with degenerate rank 0, mainly the mixed problem. We first give the formulation of the problem, and then the estimates of solutions, which can be used to prove that solutions to the above problem exist. In this article, we use the complex method, i.e., first discuss the corresponding problem for degenerate elliptic complex equations of first order, and later discuss the above problem for the degenerate elliptic equations of second order. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Discontinuous Riemann–Hilbert problems for quasilinear degenerate elliptic complex equations of first order§

In this article we discuss discontinuous Riemann–Hilbert problems for quasilinear degenerate elliptic systems of first order equations in a bounded simply connected domain. Firstly the representation of solutions of the boundary value problems for the equations is given, and then the existence and uniqueness of solutions for the problems are proved.

The mixed problem for degenerate elliptic equations of second order

The present paper deals with the mixed boundary value problem for elliptic equations with degenerate rank 0. We first give the formulation of the problem and estimates, of solutions of the problem, and then prove the existence of solutions of the above problem for elliptic equations by the above estimates and the method of parameter extension. We use the complex method, namely first discuss the corresponding problem for degenerate elliptic complex equations of first order, afterwards discuss the above problem for degenerate elliptic equations of second order.

Boundary Value Problems for Degenerate Hyperbolic Systems of First Order Equations

In this paper we discuss some boundary value problems for degenerate hyperbolic complex equations of first order in a simply connected domain, in which the boundary value problems include the Riemann-Hilbert problem and the Cauchy problem. We first give the representation of solutions of the boundary value problems for the equations, and then prove the uniqueness and existence of solutions for the problems. In [A.V. Bitsadze (1988). Some Classes of Partial Differential Equations . Gordon and Breach, New York; A.V. Bitsadze and A.N. Nakhushev (1972). Theory of degenerating hyperbolic equations. Dokl. Akad. Nauk, SSSR , 204 , 1289-1291 (Russian); M.H. Protter (1954). The Cauchy problem for a hyperbolic second order equation. Can. J. Math ., 6 , 542-553], the authors discussed some boundary value problems for hyperbolic equations of second order.

NUMERICAL SOLUTIONS OF DISCONTINUOUS BOUNDARY VALUE PROBLEMS FOR GENERAL ELLIPTIC COMPLEX EQUATIONS OF FIRST ORDER

In this paper, authors discuss the numerical methods of general discontinuous boundary value problems for elliptic complex equations of first order. They first give the well posedness of general discontinuous boundary value problems, reduce the discontinuous boundary value problems to a variation problem, and then find the numerical solutions of above problem by the finite element method. Finally authors give some error-estimates of the foregoing numerical solutions.

Some discontinuous boundary value problems for nonlinear elliptic complex equations of first order in the plane

In Refs. [1], [2] and [4], we discussed the discontinuous Riemann-hilbert boundary value problems for nonlinear uniformly elliptic complex equations of first order in a multiply connected plane domain, but, there the solution for the modified boundary value problem with the indeces K<-1 has a pole of degree in the domain. In this paper, we give a new well posedness of the discontinuous Riemann-hilbert problem for elliptic complex equation of first order, so that its solution will be continuous in the domain. One of the significances of the paper is that the present proof of solvability of the modified boundary value problem for analytic functions and generalized analytic functions is new and simpler, which is mainly based on the zero property of analytic functions and a priori estimates of solutions and some other arguments without using the method of integral equations.

The solution of riemann–hilbert boundary value problem for elliptic complex equations of first order in the sobolev space

In this paper, we give a new proper formulation of Riemann-Hilbert boundary value problem for non-linear elliptic complex equations of first order in a multiply connected domain and establish the r...

On compound boundary value problem with shift for nonlinear elliptic complex equations of first order

In this paper we are To discuss the compound boundary value problem with Haseman. Haseman type, Carleman, Carleman type boundary conditions for nonlinear uniformly elliptic complex equations of first order in the multiply connected domain. The solvability of the above compound boundary value problem, is proved by theorems of conformal gluing, the method of elimination, some a priori estimations of the solutions and the Leray-Schauder theorem. The results obtained are the generalization of the corresponding theorems in Refs. [1], [2] and [4].