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 K1（y）uxx+|K2（x）|uyy+a（x,y）ux+b（x, y）uy+c（x,y）u=-d（x,y） in any plane domain D with the boundary D=Γ∪L1∪L2∪L3∪L4, whereΓ（■{y】0}）∈Cμ2 （0【μ【1） is a curve with the end points z=-1,1. L1, L2, L3, L4 are four characteristics with the slopes -H2（x）/H1（y）, H2（x）/H1（y）,-H2（x）/H1（y）, H2（x）/H1（y）(H1（y）=|k1（y）|1/2, H2（x）=|K2（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∪L1∪L4, Im[λ（z）uz]|z=zl=bl, l=1,2, u（-1）=b0, u（1）=b3, in which z1, z2 are the intersection points of L1, L2, L3, L4 respectively. The above equations can be called the general Chaplygin-Rassias equations, which include the Chaplygin-Rassias equations K1（y）（M2（x）ux）x+M1（x）（K2（y）uy）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）uxx+uyy=0 with the boundary condition u（z）=φ（z） onΓ∪L1∪L4 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）=uz=[H1（y）ux-iH2（x）uy]/2 in the elliptic domain and W（z）=W（x+jy）=uz=[H1（y）ux-jH2（x）uy]/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.

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.