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.     

Oblique derivative problems for second order nonlinear equations of mixed type in multiply connected domains

In this paper, the unique solvability of oblique derivative boundary value problems for second order nonlinear equations of mixed (elliptic-hyperbolic) type in multiply connected domains is proved, which mainly is based on the representation of solutions for the above boundary value problem, and the uniqueness and existence of solutions of the above problem for the equation u_xx + sgn y u_yy = 0.     

Oblique derivative problems for linear mixed equations of second order

Oblique derivative boundary value problems for the linear mixed (elliptic-hyperbolic) equation of the second order, i.e. the generalized Lavrent’ev-Bitsadze equation with weak conditions are discussed. The representation of solutions for the above boundary value problem is given, the uniqueness and existence of solutions of the above problem are proved, and a priori estimates of the solutions of the above problem are obtained. Project supported by the National Natural Science Foundation of China.     

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

This paper considers the Riemann-Hilbert problem for linear mixed（elliptichyperbolic） 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.     

Solvability of the Tricomi problem for second order equations of mixed type with degenerate curve on the sides of an angle

In [1]–[6], the author posed and discussed the Tricomi problem of second order mixed equations, but he only consider some special mixed equations. In [3], the author discussed the uniqueness of solutions of the Tricomi problem for some second order mixed equation with nonsmooth degenerate line. The present paper deals with the Tricomi problem for general second order mixed equations with degenerate curve on the sides of an angle. I first give the formulation of the above problem, and then prove the solvability of the Tricomi problem for the mixed equations with degenerate curve on the sides of an angle, by using the existence of solutions of the mixed problem for the degenerate elliptic equations (see [11]). Here I mention that the used method in this paper is different to those in other papers or books, because I introduce the new notation (2.1) below, such that the second order equation of mixed type can be reduced to the first order complex equation of mixed type with singular coefficients, hence I can use the advantage of complex analytic method. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

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) |K_2(x)|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, whereΓ(■{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)|~(1/2), H_2(x)=|K_2(x)|~(1/2) in {y<0}) passing through the points z=x iy=-1,0,0,1 respectively. And the boundary condition possesses the form 1/2 u/v=1/H(x,y)Re[λuz]=r(z), z∈Γ∪L_1∪L_4, Im[λ(z)uz]|_(z=z_l)=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)=φ(z) onΓ∪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_z=[H_1(y)u_x-iH_2(x)u_y]/2 in the elliptic domain and W(z)=W(x jy)=u_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.     

Initial-irregular oblique derivative problems for nonlinear parabolic systems of second order equations with measurable coefficients

The initial-irregular oblique derivative boundary value problems for nonlinear and nondivergence parabolic systems of second order equations in multiply connected domains are dealt with where coefficients of systems of equations are meaurable. The uniqueness theorem of solutions for the above problems and somea priori estimates of solutions for the problems are given. And by using the above estimates of solutions and the Leray-Schauder theorem, the existence of solutions of the initial-boundary value problems is proved. The results are generalizations of corresponding theorems in literature. Project supported by the National Natural Science Foundation of China (Grant No. 19671006).     

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.     

Estimates of solutions of initial-irregular oblique derivative problems for fully nonlinear parabolic equations

In this paper the initial-irregular oblique derivative problems for fully nonlinear parabolic equations of second order are proposed, and then some a priori estimates of solutions for the above problems are given.     

Oblique derivative problems for nonlinear elliptic systems with measurable coefficients

1 Problem formulation Let Q be a bounded multiply connected domain in RN and the boundary OQ E C2. W6consider the nonlinear elliptic system of second order equationsUnder certain conditions, system (1) can be reduced to the formwhere u = (ul,' t u.), Da = (u..), DZu = (u:..,), andSuppose that (1) (or (2)) satisfiesCondition C For arbitrary functions u'(x), u'(z) E Cd(~) n W::(Q), Fk(x,u,Du,DZu)(k = 1,' I m) satisfy the conditionswhere 0 < g < 1, u == al - u2, and al:), bit), of*),…     

The degenerate hyperbolic problem for the two-dimensional nonlinear wave system

This paper constructs a local classical solution of degenerate hyperbolic problem for the two-dimensional nonlinear wave system. To deal with the parabolic degeneracy, we introduce a new set of coordinates to transform the nonlinear wave system to a new system that has explicitly singularity-regularity structures. By constructing a weighted metric space, we establish the existence of solution for the new system. Returning the solution to the original variables, we obtain the existence of the classical solution for the nonlinear wave system with degenerate boundary value problem.     

On the Cauchy problem of degenerate hyperbolic equations

In this paper, we study a class of degenerate hyperbolic equations and prove the existence of smooth solutions for Cauchy problems. The existence result is based on a priori estimates of Sobolev norms of solutions. Such estimates illustrate a loss of derivatives because of the degeneracy.

     

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).     

Oscillatory properties for semilinear degenerate hyperbolic equations of second order

We consider three kind of oscillatory properties of the solutions to semilinear degenerate hyperbolic equations. Several sufficient conditions for the oscillation or non-oscillation are presented. In particular, they give us the positivity of the solutions for semilinear hyperbolic equations degenerating at initial point in one space dimension. Moreover we establish a few oscillatory conditions for the solutions of the mixed problem reduced to in one space dimension.     

Multidimensional transonic shocks and free boundary problems for nonlinear equations of mixed type

We establish the existence and stability of multidimensional transonic shocks for the Euler equations for steady potential compressible fluids. The Euler equations, consisting of the conservation law of mass and the Bernoulli law for the velocity, can be written as a second-order, nonlinear equation of mixed elliptic-hyperbolic type for the velocity potential. The transonic shock problem can be formulated into the following free boundary problem: The free boundary is the location of the transonic shock which divides the two regions of smooth flow, and the equation is hyperbolic in the upstream region where the smooth perturbed flow is supersonic. We develop a nonlinear approach to deal with such a free boundary problem in order to solve the transonic shock problem. Our results indicate that there exists a unique solution of the free boundary problem such that the equation is always elliptic in the downstream region and the free boundary is smooth, provided that the hyperbolic phase is close to a uniform flow. We prove that the free boundary is stable under the steady perturbation of the hyperbolic phase. We also establish the existence and stability of multidimensional transonic shocks near spherical or circular transonic shocks.