OBLIQUE DERIVATIVE PROBLEMS FOR SECOND ORDER NONLINEAR MIXED EQUATIONS WITH DEGENERATE LINE

The present article deals with oblique derivative problems for some nonlinear mixed equations with parabolic degeneracy, which include the Tricomi problem as a special case. First, the formulation of the problems for the equations is given; next, the representation and estimates of solutions for the above problems are obtained; finally, the existence of solutions for the problems is proved by the successive iteration and the compactness principle of solutions of the problems. In this article, the author uses the complex method, namely, the complex functions in the elliptic domain and the hyperbolic complex functions in hyperbolic domain are used.

Oblique Derivative Problems for Second Order Nonlinear Equations of Mixed Type with Degenerate Hyperbolic Curve

The present paper deals with oblique derivative problems for second order nonlinear equa- tions of mixed type with degenerate hyperbolic curve, which include the Tricomi problem as a special case. Firstly the formulation of the problems for the equations is given, next the representation and estimates of solutions for the above problems are obtained, finally the existence of solutions for the problems is proved by the successive iteration of solutions of the equations and the fixed-point princi- ple. In this paper, we use the complex analytic method, namely the new partial derivative notations, elliptic complex functions in the elliptic domain and hyperbolic complex functions in the hyperbolic domain are introduced, such that the second order equations of mixed type with degenerate curve are reduced to the first order mixed complex equations with singular coefficients, and then the advantage of complex analytic method can be applied.

The oblique derivative problem for nonlinear elliptic complex equations of second order in multiply connected unbounded domains

In this article,we discuss that an oblique derivative boundary value problem for nonlinear uniformly elliptic complex equation of second order with the boundary conditions in a multiply connected unbounded domain D.The above boundary value problem will be called Problem P.Under certain conditions,by using the priori estimates of solutions and Leray-Schauder fixed point theorem,we can obtain some results of the solvability for the above boundary value problem（0.1） and（0.2）.

AN IRREGULAR OBLIQUE DERIVATIVE PROBLEM FOR SOME NONLINEAR ELLIPTIC EQUATIONS OF SECOND ORDER

This paper deals the irregular oblique derivative problem for some nonlinear elliptic equations of second order. First a priori estimates of solutions are given, afterwards by using the above estimates of solutions and the Schauder fixed-point theorem, the existence of solutions for the above boundary value problems is proved.

Boundary value problems for two types of degenerate elliptic systems in R^4

Firstly, the Riemann boundary value problem for a kind of degenerate elliptic sys- tem of the first order equations in R4 is proposed. Then, with the help of the one-to-one correspondence between the theory of Clifford valued generalized regular functions and that of the degenerate elliptic system＇s solution, the boundary value problem as stated above is trans- formed into a boundary value problem related to the generalized regular functions in Clifford analysis. Moreover, the solution of the Riemann boundary value problem for the degenerate elliptic system is explicitly described by using a kind of singular integral operator. Finally, the conditions for the existence of solutions of the oblique derivative problem for another kind of degenerate elliptic system of the first order equations in R4 are derived.

Estimates of solutions of initial-irregular oblique derivative problems for linear parabolic equations of second order with measurable coefficients

The initial\|irregular oblique derivative boundary value problems for linear and nondivergence parabolic complex equations of second order in multiply connected domains are dealt with, where the coefficients of equations are measurable. Firstly the uniqueness of solutions for the above problems is introduced, and then some \%a priori\% estimates of solutions for the problems are given. By using the above estimates and the Leray\|Schauder theorem, the existence of solutions of the initial\|boundary value problems can be proved. The results are generalizations of corresponding theorems in literature.

Oblique derivative problem for general Chaplygin-Rassias equations

The present paper deals with the oblique derivative problem for general second order equations of mixed (elliptic-hyperbolic) type with the nonsmooth parabolic degenerate line $$K_1 (y)u_{xx} + \left| {K_2 (x)} \right|u_{yy} + a(x,y)u_x + b(x,y)u_y + c(x,y)u = - d(x,y)$$ in any plane domain D with the boundary ?D=Γ ∪ L 1 ∪ L 2 ∪ L 3 ∪ L 4, where Γ(? {y > 0}) ∈ C μ 2 (0 < μ < 1) is a curve with the end points z = ?1, 1. L 1, L 2, L 3, L 4 are four characteristics with the slopes ?H 2(x)/H 1(y), H 2(x)/H 1(y),?H 2(x)/H 1(y),H 2(x)/H 1(y) (H 1(y) = √|K 1(y)|, H 2(x) = √|K 2(x)| in {y < 0}) passing through the points z = x + iy = ?1, 0, 0, 1 respectively. And the boundary condition possesses the form $$\frac{1}{2}\frac{{\partial u}}{{\partial \nu }} = \frac{1}{{H(x,y)}}\operatorname{Re} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right] = r(z), z \in \Gamma \cup L_1 \cup L_4 , \operatorname{Im} \left[ {\overline {\lambda (z)} u_{\tilde z} } \right]\left| {_{z = z_l } } \right. = b_l ,l = 1,2, u( - 1) = b_0 ,u(1) = b_3 ,$$ in which z 1, z 2 are the intersection points of L 1, L 2, L 3, L 4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations $$K_1 (y)(M_2 (x)u_x )_x + M_1 (x)(K_2 (y)u_y )_y + r(x,y)u = f(x,y), in D$$ as their special case. The above boundary value problem includes the Tricomi problem of the Chaplygin equation: K(y)u xx+u yy = 0 with the boundary condition u(z) = ?(z) on Γ ∪ L 1 ∪ L 4 as a special case. Firstly some estimates and the existence of solutions of the corresponding boundary value problems for the degenerate elliptic and hyperbolic equations of second order are discussed. Secondly, the solvability of the Tricomi problem, the oblique derivative problem and Frankl problem for the general Chaplygin-Rassias equations are proved. The used method in this paper is different from those in other papers, because the new notations W(z) = W(x + iy) = $u_{\tilde z} $ = [H 1(y)u x ? iH 2(x)u y]/2 in the elliptic domain and W(z) = W(x+jy) = $u_{\tilde z} $ =[H 1(y)u x ? jH 2(x)u y]/2 in the hyperbolic domain are introduced for the first time, such that the second order equations of mixed type can be reduced to the mixed complex equations of first order with singular coefficients. And thirdly, the advantage of complex analytic method is used, otherwise the complex analytic method cannot be applied.

Oblique Derivative Problem for Quasilinear Mixed (Lavrentév-Bitsadze) Equations of Second Order in Two Connected Domains

In this article, we first give the representation of solutions for the oblique derivative problem of mixed （Lavrentév-Bitsadze） equations in two connected domains, afterwards prove the uniqueness of solutions of the above problem. Moreover, we prove the solvability of oblique derivative problem for quasilinear mixed （Lavrentév-Bitsadze） equations of second order, and obtain a priori estimates of solutions of the above problem. The above problem is an open problem proposed by Rassias.

Integral Equation Method for Inverse Scattering Problem from the Far-Field Data

Consider the inverse scattering problem in terms of Helmholtz equation.We study a simply connected domain with oblique derivative boundary condition.In the case of constant l,given a finite number of incident wave,the shape of the scatterer is reconstructed from the measured far-field data.We propose a Newton method which is based on the nonlinear boundary integral equation.After computing the Fr´echet derivatives with respect to the unknown boundary,the nonlinear equation is transformed to its linear form,then we show the iteration scheme for the inverse problem.We conclude our paper by presenting several numerical examples for shape reconstruction to show the validity of the method we presented.