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) + vertical bar K-2(x)vertical bar 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. partial derivative D= Gamma boolean OR L-1 boolean OR L-2 boolean OR L-3 boolean OR L-4, where Gamma(subset of{y > 0}) is an element of C-mu(2) (0 < mu < 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) = root vertical bar K-1(y)vertical bar, H-2(x) = root vertical bar K-2(x)vertical bar in {y < 0}) passing through the points z = x + iy = - 1, 0, 0, 1 respectively. And the boundary condition 1/2 partial derivative u/partial derivative v = 1/H(x,y)Re<(lambda(z))over bar>(z)u((z) over bar)]=r(z), z is an element of Gamma boolean OR L1 boolean OR L4, Im[< (lambda(z))over ba r>(z)u((z) over bar)]vertical bar(z=zl)=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) = f(z) phi(z) on Gamma boolean OR L-1 boolean OR 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 Frank1 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) over bar) = [H-1(y)u(x) - iH(2)(x)u(y)]/ 2 in the elliptic domain and W(z)= W(x+ jy)= u((z) over bar) =[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.