THE GENERALIZED RIEMANN-HILBERT PROBLEM FAR A MULTI-CONNECTED REGION OF SECOND ORDER NON-LINEAR ELLIPTIC EQUATIONS

In this paper, we consider the generalized Riemann-Hilbert problem for second order non-linear elliptic complex equation(?),z∈G (1) with the boundary conditionRe[z~(n_1)e~(-∏ia_1(z))w]=r_1(z), Re[z~(-n_2)e~(∏ia_2(z))(?)]=r_2(z), z∈Γ, (2) where Γ=Γ_0+Γ_1+…+Γ_m is the smooth boundary of a multi-connected region G, n_i(i=1, 2) are called the indices of the boundary value problem.we also obtain the following existence theorem of generalized solution.Theorem. suppose that the indices n_im-1, the coefficients of the complex equation (1) and the boundary condition (2) satisftes the condition (c), and q~0 is sufficien tly small, then the seneralized Riemann-Hilbert problem (1), (2) is solvable and the solution has theexpression (7).